Realizability and automatic realizability of Galois groups of order 32

DOI: 10.2478/s11533-009-0072-x

Central European Journal of Mathematics

Realizability and automatic realizabilityof Galois groups of order 32

Research Article

Helen G. Grundman1∗, Tara L. Smith 2†

1 Bryn Mawr College, Bryn Mawr, PA 19010, USA

2 University of Cincinnati, Cincinnati, OH 45221, USA

Received 16 October 2009; accepted 2 December 2009

Abstract: This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galoisgroup over an arbitrary field. These conditions, given in terms of the number of square classes of the field andthe triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizabilityresults.

MSC: 12F12

Keywords: Inverse Galois theory • 2-groups • Automatic realizability

© Versita Sp. z o.o.

1. Introduction

The inverse problem of Galois theory for a field K asks which finite groups can be realized as Galois groups Gal(L/K )for some Galois extension L of K or, alternatively, for a finite group G, which fields K have a Galois extension L such thatGal(L/K ) ∼= G. The problem is solved for fields of characteristic 2 by a classic result of Witt [18]. Therefore throughoutthis paper we assume all fields are of characteristic not 2.Extensive research has focused on small 2-groups. A compilation of conditions for the realizability of groups of order 2,4, 8, or 16 over fields of characteristic not 2, as well as descriptions of field extensions with those Galois groups, can befound in [6] and [12]. In this work, we give necessary and sufficient conditions for the 51 groups of order 32, computingsome directly, but also drawing together results spread throughout the literature. In Section 3, we use these conditionsto prove automatic realizability results concerning these groups.In general, there is no universally accepted notation for the groups of order 32. The most common numbering schemesfor small 2-groups can be found in [3] and in [8]. We use G(r,s) to denote the group of order 32 with Hall-Senior numberr (see Hall and Senior [8]) and [32, s] in GAP’s small groups catalog (see [3]). For appropriate values of n, we let Cn, Dn,

∗ E-mail: [email protected]† E-mail: [email protected]

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Cent. Eur. J. Math. • 8(2) • 2010 • 244-260

H.G. Grundman, T.L. Smith

SDn, Qn and Mn denote the cyclic, dihedral, semidihedral, quaternion, and modular groups of order n. For completeness,we give group presentations of the nonabelian groups of order 32 that are not direct products of smaller groups at theend of the paper.We first consider realizability conditions for the abelian groups of order 32 and nonabelian groups of order 32 that canbe expressed as direct products of smaller groups. By a well-known result of Kuyk and Lenstra [11], C32 is realizable asa Galois group over a field K if and only if C4 is realizable and thus if and only if there exists a nonsquare a ∈ K̇ with(a,−1) = 1 in the Brauer group of K . For the rest of these groups, we note that a necessary and sufficient conditionfor the realizability of the product of two finite 2-groups over a field K is the realizability of the two groups over K byfields that have no common quadratic subextension of K . So the desired realizability conditions are easily derived fromthose of smaller 2-groups.In Table 1, we list necessary and sufficient realizability conditions for the abelian groups of order 32. The first columndescribes the group in the notation introduced earlier and as a product of cyclic groups; the second column lists requiredquadratically independent elements in K̇ ; and the third column lists elements of Br(K ), written multiplicatively, that arerequired to be trivial for the group to be realizable over K .

Table 1. Abelian Groups

Group Quad ind in K̇ Trivial elements in Br(K )

G(1,51)∼= (C2)5 a, b, c, d, e

G(2,45)∼= (C2)3 × C4 a, b, c, d (a,−1)

G(3,21)∼= C2 × (C4)2 a, b, c (a,−1), (b,−1)

G(4,36)∼= (C2)2 × C8 a, b, c (a,−1)

G(5,3)∼= C4 × C8 a, b (a,−1), (b,−1)

G(6,16)∼= C2 × C16 a, b (a,−1)

G(7,1)∼= C32 a (a,−1)

In Table 2, we provide necessary and sufficient realizability conditions for the nonabelian direct products of order 32,in the same format as Table 1. In this table, some of the elements of Br(K ) involve an additional unrestricted elementx ∈ K̇ .

Table 2. Nonabelian Direct Products


G(8,46)∼= (C2)2 ×D8 a, b, c, d (a,−b)

G(9,47)∼= (C2)2 ×Q8 a, b, c, d (a,−b)(b,−1)

G(10,48)∼= C2 ×DC a, b, c, d (a,−b)(c,−1)

G(11,22)∼= C2 × (D8 f C4) a, b, c (a,−1), (a,−b)

G(12,23)∼= C2 × (C4 o C4) a, b, c (a,−b), (b,−1)

G(13,37)∼= C2 ×M16 a, b, c (a,−1), (a, 2b)(x,−1)

G(14,25)∼= C4 ×D8 a, b, c (a,−1), (b,−c)

G(15,26)∼= C4 ×Q8 a, b, c (a,−1), (b,−c)(c,−1)

G(23,39)∼= C2 ×D16 a, b, c (a,−b), (ab, 2)(−b, x)

G(24,40)∼= C2 × SD16 a, b, c (a,−b), (a,−2)(−b, x)

G(25,41)∼= C2 ×Q16 a, b, c (a,−b), (ab, 2)(b,−1)(−b, x)

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Realizability and automatic realizability of Galois groups of order 32

2. Realizability of non-direct products

In this section, we discuss the nonabelian groups of order 32 that cannot be decomposed into direct products of smallergroups. Realizability conditions for some of these groups can be found in the literature. For others, the conditions canbe derived from results concerning related embedding problems, as we now describe.Given a Galois extension L/K and a short exact sequence

1 −−−−−→ N −−−−−→ι

G −−−−−→ψ

Gal(L/K ) −−−−−→ 1, (1)

the Galois embedding problem of L/K and (1) asks if there exist a Galois field extension M/K with L ⊂ M and ahomomorphism φ :Gal(M/K ) → G such that ψφ :Gal(M/K ) → Gal(L/K ) is the natural restriction of Galois groups. Thepair (M/K, φ) is a proper solution if φ is a surjection. It is easy to see that the group G is realizable over K if and onlyif G/ι(N) is realizable over K , say by a field L, and the Galois embedding problem given by L/K and (1) is properlysolvable.We note that if the exact sequence is not split and has kernel N ∼= C2, then any solution is necessarily proper.Realizability conditions for the 13 groups of order 32 that are extensions of (C2)r × (C4)s × (D4)t by C2 were determinedby Grundman and Stewart [7]. We give these in Table 3, using the same format as the earlier tables. (In the conditionsin Table 3, x and y represent unrestricted elements in K̇ .)If the kernel is N1 ×N2

∼= C2 × C2, then the embedding problem given by L/K and (1) is properly solvable if and onlyif each of the problems given by L/K and

1 −−−−−→ N1 −−−−−→ι

G/N2 −−−−−→ψ

Gal(L/K ) −−−−−→ 1, (2)

1 −−−−−→ N2 −−−−−→ι

G/N1 −−−−−→ψ

Gal(L/K ) −−−−−→ 1, (3)

is properly solvable [9]. Thus, unless one of these sequences splits, again, any solution is necessarily proper. Wecombine this observation with the methods used by Ledet [12] and by Grundman and Stewart [7] to determine realizabilityconditions for the remaining nine groups that are extensions of (C2)r × (C4)s × (D4)t by C2 × C2. We give these resultsin Table 3 as well. As an example of the derivation of these new realizability conditions, we prove the condition for thegroup G(21,12).

Theorem 2.1.The group

G(21,12) = 〈u, v|u4 = v8 = 1; v−1uv = u3〉

is realizable as a Galois group over a field K if and only if there exist quadratically independent elements a and b ∈ K̇and x ∈ K̇ such that (a,−1) = (−a, b) = (a, 2)(x,−1) = 1 in Br(K ).

Proof. The group C2 ×C4 is realizable over K if and only if there exist quadratically independent elements a1, a2 ∈ K̇such that (a2,−1) = 1. Let L/K be a C2 × C4-extension, if one exists.Letting σ1 of order 2 and σ2 of order 4 generate C2 × C4, consider the exact sequence

1 −−−−−→ C2 × C2 −−−−−→(−1,1) 7→u2(1,−1) 7→v4

G(21,12) −−−−−→u 7→σ1v 7→σ2

C2 × C4 −−−−−→ 1.(4)

The embedding problem given by L/K and (4) is solvable if and only if the embedding problems given by L/K and eachof the sequences

1 −−−−−→ C2 −−−−−→−17→v4

G(21,12)/〈u2〉 −−−−−→u 7→σ1v 7→σ2

C2 × C4 −−−−−→ 1 (5)

and1 −−−−−→ C2 −−−−−→

−17→u2G(53,97)/〈v4〉 −−−−−→

u 7→σ1v 7→σ2

C2 × C4 −−−−−→ 1. (6)

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Table 3. Non-Direct Products


G(16,24) a, b, c (a,−1)(b, c), (c,−1)G(17,38) a, b, c (a, b)(c, 2)(x,−1), (c,−1)G(18,2) a, b (a, b), (a,−1), (b,−1)G(19,4) a, b (a,−1), (b,−1), (a, 2b)(x,−1)G(20,5) a, b (a,−1), (a, b), (a, 2)(x,−1)G(21,12) a, b (a,−1), (−a, b), (a, 2)(x,−1)G(26,42) a, b, c (a, c)(b, 2)(−c, x), (b,−c)G(27,9) a, b (a,−b), (a,−1), (a, 2)(−b, x)G(28,10) a, b (a,−b), (ab,−1), (a,−2)(−b, x)G(29,14) a, b (a,−b), (b,−1), (a, 2)(−b, x)G(30,13) a, b (a,−b), (b,−1), (a,−2)(−b, x)G(31,11) a, b (a, b), (b,−1)G(33,27) a, b, c (a, bc), (b,−c)G(34,34) a, b, c (a,−c), (b,−c)G(35,35) a, b, c (a,−c)(c,−1), (b,−c)G(36,28) a, b, c (a, c), (b,−c)G(37,29) a, b, c (a,−1)(bc,−a), (b,−c)G(38,30) a, b, c (b,−a)(c,−1), (b,−c)G(39,31) a, b, c (a,−c)(b,−1), (b,−c)G(40,32) a, b, c (a, b)(ac,−1), (a, c)(b,−1)G(41,33) a, b, c (a, b)(bc,−1), (a, c)(b,−1)G(42,49) a, b, c, d (a, b)(c, d)G(43,50) a, b, c, d (a,−b)(b,−1)(c, d)G(44,43) a, b, c (b, 2a)(−c, x), (b,−c)G(45,44) a, b, c (b, 2a)(−c, x)(c,−1), (b,−c)G(46,6) a, b (a,−1), (a, b)G(47,7) a, b (a, b), (b,−1)G(48,8) a, b (a, b), (b,−1), (−a, x)(b, y)(−1,−1)

are solvable.Using the notation and results of [7, Theorem 1], we choose preimages g1 = u of σ1 and g2 = v of σ2. Then forsequence (5), d2 = 1 and d1 = d12 = 0, yielding the obstruction (a2, 2)(−1, x) with x unrestricted in K̇ . For sequence (6),d1 = d12 = 1 and d2 = 0, yielding the obstruction (a1,−1)(a1, a2). The theorem follows.

Also in Table 3 are the realizability conditions for G(42,49) and G(43,50) determined by Smith [16] and for G(46,6) determinedby Gao, Leep, Minác, and Smith [2], and realizability conditions for G(31,11), G(47,7), and G(48,8) that follow from embeddingobstructions determined by Swallow and Thiem [17] and the realizability condition for D8 fC4, which can be found in [6].(For some of the above groups, relevant obstructions have also been computed in [15] and for others, in [17].)There are five remaining groups of order 32. The solutions to the embedding problems corresponding to these groupsare much more complicated, involving the Brauer groups of at least one extension field along with the Brauer groupof K . These results are given below in Theorems 2.2, 2.4, 2.5, 2.6, and 2.7. (In each of these theorems, we follow thenotational convention that for any nonzero a, (a2, 0) = (0, a2) = 1 in the Brauer group.)The necessary and sufficient conditions for the realizability of G(32,15) were determined by Grundman and Smith [5], asdescribed in the following theorem.

Theorem 2.2.Let K be a field. The group G(32,15) is realizable over K if and only if either

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1. i ∈ K and there exist a, b, x ∈ K̇ with a and b quadratically independent such that (a, b) = (a, 2)(b, x) = 1 inBr(K ) or

2. i /∈ K and at least one of the following holds:

• there exist a, b, x ∈ K̇ such that a, b, and −1 are quadratically independent, (a,−b) = (b,−1) = 1 inBr(K ), and (a, 2)(b, x) = 1 in Br(K (i)); or

• there exist b, x ∈ K̇ such that b and −1 are quadratically independent, (b,−1) = (−1,−1) = 1 in Br(K )and (b, x(αi− β)) = 1 in Br(K (i)), where α, β ∈ K with α2 + β2 = −b; or

• there exists a ∈ K̇ such that a and −1 are quadratically independent, (−1,−1) = 1 in Br(K ) and (a, 2) = 1in Br(K (i)); or

• there exist a, x ∈ K̇ such that a and −1 are quadratically independent, (a,−1) = (−1,−1) = 1 in Br(K ),and (a, x(αi− a)) = 1 in Br(K (i)) where α, β ∈ K with α2 + a2 = aβ2.

To derive realizability conditions for M32 = G(22,17), we use the following obstructions determined by Michailov [14,Theorem 3.3].

Theorem 2.3 (Michailov).Let ζ be a primitive 2nth root of unity, with n ≥ 2, such that ζ+ζ−1 ∈ K and (ζ−ζ−1)/i ∈ K . Let L = K (

√r(a+

√a),

√b)

be a C4 × C2 extension of K with a = 1 + c2, b, r ∈ K̇ . Then the embedding problem given by L/K and the extension

1 −−−−−→ C2n −−−−−→ M2n+3 −−−−−→ C4 × C2 −−−−−→ 1, (7)

has the following obstructions:

1. If i ∈ K , (a, bα)(ζ, rαβ) ∈ Br(K ), where α ∈ K̇ and β ∈ K , such that α2 − aβ2 = ζ.

2. If a = −1, (−1, r) ∈ Br(K ) and (r(1 − iu), ζ) ∈ Br(K (i)), where u2 + v2 = −1 with u, v ∈ K and L =K (

√r(1 − iu),

√b).

3. If b = −1 or ab = −1, (a, 2)(−1, r) ∈ Br(K ) and (a, α)(ζ, r(1 − ic)αβ) ∈ Br(K (i)), where α ∈ ˙K (i) and β ∈ K (i),such that α2 − aβ2 = ζ.

4. If a, b, and −1 are quadratically independent, (a, 2)(−1, r) ∈ Br(K ) and (a, αb)(ζ, r(1 − ic)αβ) ∈ Br(K (i)) whereα ∈ ˙K (i) and β ∈ K (i), such that α2 − aβ2 = ζ.

The following lemma allows us to utilize these obstructions to determine realizability conditions for M32.

Lemma 2.1.Let L be a C4 × C2 extension of a field K . Any solution to the embedding problem of L/K and

1 −−−−−→ C4 −−−−−→ M32 −−−−−→ψ

C4 × C2 −−−−−→ 1 (8)

is necessarily a proper solution.

Proof. First note that since M32 has exponent 16, the sequence is not split. Suppose (M/K, φ) is a solution to theembedding problem that is not proper. Without loss of generality, we may assume that φ is injective. Then φ is notsurjective and thus its image, which is isomorphic to Gal(M/K ), is isomorphic to a subgroup of order 16 in G that hasC4 ×C2 as a quotient group. Examining the possibilities in [1], we see that Gal(M/K ) ∼= C8 ×C2. Let z be an element oforder 8 in Gal(M/K ) ∼= C8 × C2. Under the natural projection to Gal(L/K ) ∼= C4 × C2, z maps to an element of order 4.Since φ is injective, φ(z) is an element of M32 of order 8. Let M32 be given by the presentation 〈x, y|x16 = y2 = 1; [y, x] =x8〉. Then φ(z) maps to an element of the form x2a or x2ay with a odd. Now, ψ(x) is of order at most 4 and so ψ(x2a)is of order of at most 2. Further, ψ(x2ay) = ψ(x2a)ψ(y), which is a product of two elements of C4 × C2 each of order atmost 2. But this means that ψ(x2ay) is of order at most 2. So, in any case, the order of ψ(φ(z)) is not 4 and thereforeψφ does not equal the natural projection of Galois groups. Hence any solution to the embedding problem L/K and (8)is a proper solution.

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The realizability conditions for M32 now follow from the well-known realizability condition for C2 × C4.

Theorem 2.4.The group M32 = G(22,17) is realizable over K if and only if there exist quadratically independent elements a, b ∈ K̇ andx, y ∈ K̇ such that a = 1 + y2 and there exist α ∈ K̇ (i) and β ∈ K (i) such that i = α2 − aβ2 and any of the followingholds:

1. i ∈ K and (a, bα)(i, xαβ) = 1 in Br(K ).

2. −a ∈ K̇ 2, say −a = w2 with w ∈ K̇ , (x,−1) = 1 in Br(K ) and (x(1 − iw), i) = 1 in Br(K (i)).

3. −b or −ab ∈ K̇ 2, (a, 2)(−1, x) = 1 in Br(K ), and (a, α)(i, x(1 − iy)αβ) = 1 in Br(K (i)).

4. a, b, and −1 are quadratically independent, (a, 2)(−1, x) = 1 in Br(K ) and (a, αb)(i, x(1 − iy)αβ) = 1 in Br(K (i)).

Proof. As noted at the beginning of this section, M32 is realizable over K if and only if M32/ι(C4) is realizable over K ,say by a field L, and the Galois embedding problem given by L/K and (9) is properly solvable. Now, M32/ι(C4) ∼= C4 ×C2

is realizable over K if and only if there exist elements a, b, x, and y ∈ K̇ , with a and b quadratically independent, suchthat a = 1+y2. In this case, L = K (

√x(a+

√a),

√b) is a C4 ×C2 extension of K for any x ∈ K̇ . (See, for example, [6].)

Further, by Lemma 2.1, any solution to L/K and (9) is a proper solution.Letting n = 2, the hypotheses of Theorem 2.3 are trivially satisfied with ζ = i. Thus the theorem yields obstructions forthe embedding problem of L/K and

1 −−−−−→ C4 −−−−−→ι

M32 −−−−−→ C4 × C2 −−−−−→ 1. (9)

Notice that if −a ∈ K̇ 2, say −a = w2 with w ∈ K̇ , then ( 1+i2 )2 − a( 1+i

2w )2 = i. And in any of the other cases ofTheorem 2.3, the conditions require that there exist α ∈ K̇ (i) and β ∈ K (i) such that i = α2 − aβ2. Hence, in any case,for there to be a solution to the embedding problem, such α and β must exist.The realizability conditions for each case now follow directly from the obstructions in the corresponding cases ofTheorem 2.3.

In order to derive necessary and sufficient realizability conditions for the groups D32, SD32, and Q32, we need thefollowing lemma.

Lemma 2.2.Let L be a D8 extension of a field K . For G any one of D32, SD32, and Q32, any solution to the embedding problem ofL/K and

1 −−−−−→ C4 −−−−−→ι

G −−−−−→ψ

D8 −−−−−→ 1 (10)

is necessarily a proper solution.

Proof. First note that since each of the groups D32, SD32, and Q32 has exponent 16, the sequence (10) is not split.Suppose (M/K, φ) is a solution to the embedding problem L/K and (10) that is not proper. Without loss of generality,we may assume that φ is injective. Then φ is not surjective and thus its image, which is isomorphic to Gal(M/K ), isisomorphic to a subgroup of order 16 in G that has D8 as a quotient group. Examining the possibilities in [1], we seethat Gal(M/K ) ∼= D16 or Q16. Letting σ be an element of order 8 in Gal(M/K ), φ(σ ) is of order 8 and hence is a squarein G. But then ψφ(σ ) must be of order 2 in D8, while the unique surjections from D16 and Q16 to D8 take elements oforder 8 to elements of order 4. This is a contradiction. Thus any solution to the embedding problem L/K and (10) is aproper solution.

The realizability conditions for D32, SD32, and Q32 now follow easily from the obstructions found by Ledet [13] for thethree embedding problems described in Lemma 2.2.

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Theorem 2.5.The group D32 = G(49,18) is realizable over K if and only if there exist quadratically independent elements a, b ∈ K̇and x ∈ K̇ such that (a,−b) = (2, ab)(−b, xα) = 1 in Br(K ), where α , β ∈ K satisfy α2 − aβ2 = ab, and any of thefollowing holds:

1. i ∈ K and for α ′, β′ ∈ K (√b) with α ′ 6= 0 and α ′2 − aβ′2 = i, (a, α ′)(i, xα ′β′(β − i

√b)) = 1 in Br(K (

√b)).

2. −a ∈ K̇ 2 and (2, x(α + i√b)) = 1 in Br(K (i

√b)).

3. b = −1 and for α ′, β′ ∈ K with α ′ 6= 0 and α ′2 + aβ′2 = 2, (2, α ′β′)(a, α ′(α ′ − 1)) = 1 in Br(K ).

4. −ab ∈ K̇ 2 and (2, x(α + β√a)) = 1 in Br(K (

√a)).

5. a, b, and −1 are quadratically independent and for α ′, β′ ∈ K (i√b) with α ′ 6= 0 and α ′2 + aβ′2 = 2, (a, α ′(α ′ −

1))(2, xα ′β′(β − i√b)) = 1 in Br(K (i

√b)).

Proof. Using Lemma 2.2, D32 is realizable over K if and only if D8 is realizable over K , say by a field L, and theGalois embedding problem given by L/K and (10), with G = D32, is solvable. The group D8 is realizable over K if andonly if there exist quadratically independent elements a, b ∈ K̇ such that (a,−b) = 1 in Br(K ), which implies that thereexist α , β ∈ K such that α2 − aβ2 = ab. The theorem now follows immediately from the obstructions determined byLedet [13, Theorem 3.6].

Theorem 2.6.The group SD32 = G(50,19) is realizable over K if and only if there exist quadratically independent elements a, b ∈ K̇and x ∈ K̇ such that (a,−b) = (2, ab)(−b, xα) = 1 in Br(K ), where α , β ∈ K satisfy α2 − aβ2 = ab, and any of thefollowing holds:


√b)) = 1 in Br(K (

√b)).

2. −a ∈ K̇ 2 and (2, x(α + i√b))(−1,−1) = 1 in Br(K (i

√b)).

3. b = −1 and for α ′, β′ ∈ K with α ′ 6= 0 and α ′2 + aβ′2 = 2, (2, α ′β′)(a,−α ′(α ′ − 1)) = 1 in Br(K ).

4. −ab ∈ K̇ 2 and (2, x(α + β√a)) = 1 in Br(K (

√a)).

5. a, b, and −1 are quadratically independent and for α ′, β′ ∈ K (i√b) with α ′ 6= 0 and α ′2 +aβ′2 = 2, (a,−α ′(α ′ −

1))(2, xα ′β′(β − i√b)) = 1 in Br(K (i

√b)).

Proof. Again using Lemma 2.2, SD32 is realizable over K if and only if D8 is realizable over K , say by a field L,and the Galois embedding problem given by L/K and (10), with G = SD32, is solvable. Noting that the semidihedralgroup is the same as the quasi-dihedral group, the theorem now follows from the obstructions determined by Ledet [13,Theorem 3.7].

Theorem 2.7.The group Q32 = G(51,20) is realizable over K if and only if there exist quadratically independent elements a, b ∈ K̇and x ∈ K̇ such that (a,−b) = (2, ab)(−b, xα) = 1 in Br(K ), where α , β ∈ K satisfy α2 − aβ2 = ab, and any of thefollowing holds:


√b)) = 1 in Br(K (

√b)).

2. −a ∈ K̇ 2 and (2, x(α + i√b))(−1,−1) = 1 in Br(K (i

√b)).

3. b = −1 and for α ′, β′ ∈ K with α ′ 6= 0 and α ′2 + aβ′2 = 2, (2, α ′β′)(a, α ′(α ′ − 1))(−1,−1) = 1 in Br(K ).

4. −ab ∈ K̇ 2 and (2, x(α + β√a))(−1,−1) = 1 in Br(K (

√a)).

5. a, b, and −1 are quadratically independent and for α ′, β′ ∈ K (i√b) with α ′ 6= 0 and α ′2 + aβ′2 = 2, (a, α ′(α ′ −

1))(2, xα ′β′(β − i√b))(−1,−1) = 1 in Br(K (i

√b)).

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Proof. As in the previous two proofs, we have Q32 is realizable over K if and only if D8 is realizable over K , sayby a field L, and the Galois embedding problem given by L/K and (10), with G = Q32, is solvable. Again, Ledet [13,Theorem 3.8] provides the needed obstructions from which the theorem follows immediately.

3. Automatic realizability

Automatic realizability is concerned with the question of when the realizability of one group implies the realizability ofanother group. In this section we discuss implications of the realizability conditions given in this paper. An extensivesurvey of results for 2-groups of order less than 32 can be found in [4], and from these results it is easy to derive results forgroups of order 32 (and higher) that are direct products of smaller groups. Our focus, therefore, is on results concerningthe nonabelian groups of order 32 that are not products of smaller groups. Other articles that have considered automaticrealizability questions include [15] on selected groups of order 32 and [10] on results for finite 2-groups in general.Throughout this section we assume |K̇ /K̇ 2| ≥ 8, although some of the realizability results also hold when |K̇ /K̇ 2| = 4.Many of the results depend on assumptions about the level s(K ) of the field K . Recall that s(K ) is defined to be theleast positive integer n such that −1 can be expressed as a sum of n squares, and s(K ) = ∞, if no such n exists. If s(K )is finite, then it is a power of 2.

Theorem 3.1.Realizability of any one of the groups D8fC4, G(31,11), G(46,6), or G(47,7) implies the realizability of the others. Furthermore,realizability of all of these groups is implied by realizability of any of G(18,2), G(20,5), G(27,9), G(28,10), or G(48,8).

Proof. Each of the groups in the first statement has the same realizability criteria, (a, b) = (a,−1) = 1 for quadrat-ically independent elements a, b ∈ K̇ . Each group in the second statement has D8 f C4 as a quotient.

Theorem 3.2.Assume

√−1 /∈ K . Then the group G(34,34) is always realizable over K . If also K has a C4-extension L with

√−1 /∈ L,

then the groups G(20,5), G(27,9), G(31,11), G(36,28), G(39,31), G(46,6), and G(47,7) are all realizable over K .

Proof. Letting c = −1, the realizability condition for G(34,34) is trivially satisfied. If a C4-extension L does not contain√−1, we must have (z,−1) = 1 for some z /∈ ±K̇ 2. The realizability of G(36,28) is immediate, letting a = z and c = −1.

Noting that s(K ) 6= 1, assume that s(K ) = 2 and therefore (−1,−1) = 1. Then G(20,5) and G(27,9) are realizable bychoosing a = −1, b = z, and x = 1, and G(39,31) is realizable by choosing b = z and c = −1.Now assume that s(K ) ≥ 4. First suppose

√2 /∈ K . Then since (2,−1) = 1, G(20,5) and G(27,9) are seen to be realizable

by choosing a = 2, b = −1, and x = 2, while the realizability of G(39,31) is obtained by choosing b = 2 and c = −1. If,on the other hand,

√2 ∈ K , then G(20,5) and G(27,9) can be realized by choosing a = z, b = −1, and x = 1, and G(39,31)

by choosing c = −1, and b = z.Finally, Theorem 3.1 implies the realizability of the groups G(31,11), G(46,6), and G(47,7).

Assume now that s(K ) ≥ 4. With additional assumptions on whether√

2 is in K , or on the realizability of certain C4-or D8-extensions, a number of automatic realizability results are obtained.

Theorem 3.3.Assume s(K ) ≥ 4 and suppose

√2 /∈ K . Then the groups G(16,24), G(17,38), G(20,5), G(27,9), G(31,11), G(36,28), G(39,31), G(44,43),

G(46,6), and G(47,7) are all realizable over K . If also K has a C4-extension L with√

2 /∈ L, then the groups G(19,4) andG(26,42) are realizable over K .

Proof. For G(16,24), choose b = −1 and c = 2; for G(17,38), choose a = −1, c = 2, and x = b; and for G(44,43), choosea = 2 and c = −1. The remaining groups are handled by the Theorem 3.2, since (2,−1) = 1 implies the existence of aC4-extension containing K (

√2) as its unique quadratic intermediate field.

Now suppose there exists a C4-extension L with√

2 /∈ L. Let (z,−1) = 1 be an element of Br(K ) guaranteeing theexistence of L. To realize G(19,4), choose a = z, b = 2, and x = 1. For G(26,42), choose a = z, b = 2, and c = −1.

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Theorem 3.4.Assume s(K ) ≥ 4 and suppose K has a D8-extension M with

√−1 /∈ M. Then the group G(33,27) is realizable over K . If

in addition√

2 /∈ M, then the group G(17,38) is realizable over K . If√

2 ∈ M \ K , then the group G(26,42) is realizable. If√2 ∈ K , then the group G(44,43) is realizable.

Proof. Let (z, w) = 1 be an element of Br(K ) guaranteeing the existence of the D8-extension. To realize G(33,27),choose a = z, b = −w, and c = −1. If

√2 /∈ M, G(17,38) is realized by choosing a = z, b = w, c = 2, and x = 1. If√

2 ∈ M \ K , then z, w, 2 are not quadratically independent. We may assume then that either z = 2 or zw = 2. Ifz = 2, choosing a = 2, b = w, and c = −1 realizes G(26,42). If zw = 2, we have (z, 2z) = 1 or, equivalently, (z,−2) = 1.Choosing a = z, b = 2z, and c = −1 gives the condition (z,−1)(2z, 2) = 1, or (z,−1)(2,−1)(z, 2) = (z,−2) = 1, againshowing the realizability of G(26,42). Finally, if

√2 ∈ K , then choosing a = z, b = w, and c = −1 yields G(44,43).

Theorem 3.5.Assume s(K ) ≥ 4. If

√2 ∈ K and any of the groups D8 f C4, G(31,11), G(46,6), or G(47,7) is realizable, then so are the

groups G(20,5) and G(27,9).

Proof. Let (z,−1) = (z, w) = 1 be elements of Br(K ) guaranteeing the realizability of D8 f C4 and the other threegroups. Then G(20,5) and G(27,9) are both realizable by choosing a = z and b = w.

Theorem 3.6.Assume s(K ) ≥ 4. Realizability of the group Q8 f C4 implies the realizability of G(38,30). If, in addition,

√2 ∈ K , then

the groups G(21,12), G(29,14), G(20,5), G(27,9), G(31,11), G(46,6), G(47,7), and D8 f C4 are also realizable.

Proof. Let (z,−1) = (w,−z) = 1 be elements of Br(K ) guaranteeing the existence of the Q8 f C4-extension. Sinces(K ) ≥ 4, (−1,−1) 6= 1, and so z, w, and −1 are quadratically independent in K̇ . To realize G(38,30), choose a = −1,b = w, and c = z.Now assume

√2 ∈ K . For G(21,12), choose a = z, b = w, and x = 1, and for G(29,14), choose a = w, b = z, and

x = 1. Finally observe that by [4, Theorem 3.3], the realizability of Q8 f C4 implies the realizability of D8 f C4 whens(K ) ≥ 4 and

√2 ∈ K . This implies the realizability of G(20,5) and G(27,9) by Theorem 3.5 and of G(31,11), G(46,6), G(47,7) by

Theorem 3.1.

The above results for fields of level greater than or equal to four are summarized in Table 4. For each group, G(m,n), listedin the first column of the table, the second column lists groups whose realizability implies its realizability, sometimeswith additional conditions that must be fulfilled for the automatic realizability results to hold. These conditions involvethe presence or absence of

√2 in K , or the existence of a C4-extension L over K , in some cases not containing

√2, or

the existence of a D8-extension M over K not containing√

−1 and in some cases containing√

2. The third column listsgroups whose (nontrivial) realizability is guaranteed by the realizability of the given group. It is assumed throughoutthat K contains at least as many independent square classes as the rank of the group under consideration. (Groups forwhich no easily stated automatic realizability results have been obtained have been excluded from the table.)If we assume that s(K ) = 2, then (−1,−1) = 1. Realizability of C4- or D8-extensions not containing

√−1 give additional

automatic realizability results.

Theorem 3.7.Assume s(K ) = 2. Then the groups G(21,12), G(34,34), and G(35,35) are always realizable.

Proof. The condition for G(21,12) is satisfied by letting a = −1 and x = 1, and for G(35,35) by letting c = −1. Theresult for G(34,34) follows from Theorem 3.2.

Theorem 3.8.Assume s(K ) = 2, and suppose K has a C4-extension L with

√−1 /∈ L. Then the groups G(16,24), G(18,2), G(19,4), G(20,5),

G(27,9), G(28,10), G(29,14), G(30,13), G(31,11), G(32,15), G(36,28), G(39,31), G(46,6), G(47,7), and G(48,8) are all realizable over K .

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Table 4. Realizability Results Over Fields of Level at Least Four

Group G(m,n) is realizable if G(m,n) implies

G(16,24)√

2 /∈ KG(17,38)

√2 /∈ K

G(18,2) G(31,11), G(46,6), G(47,7)

G(19,4)√

2 /∈ LG(20,5)

√2 /∈ K , C4, G(31,11), G(46,6), G(31,11), G(46,6), G(47,7)

G(47,7), Q8 f C4, or D8 f C4

G(21,12) both√

2 ∈ K and Q8 f C4

G(26,42)√

2 /∈ L or both√

2 ∈ M \ K and√

−1 /∈ MG(27,9)

√2 /∈ K , C4, G(31,11), G(46,6), G(31,11), G(46,6), G(47,7)

G(47,7), Q8 f C4, or D8 f C4

G(28,10) G(31,11), G(46,6), G(47,7)

G(29,14) both√

2 ∈ K and Q8 f C4

G(31,11)√

2 /∈ K , C4, G(18,2), G(20,5), G(46,6), G(47,7)

G(27,9), G(28,10), G(48,8), G(46,6),G(47,7), Q8 f C4, or D8 f C4

G(33,27)√

−1 /∈ MG(34,34) alwaysG(36,28)

√2 /∈ K or C4

G(38,30) Q8 f C4

G(39,31)√

2 /∈ K or C4

G(44,43)√

2 /∈ K or√

−1 /∈ MG(46,6)

√2 /∈ K , C4, G(18,2), G(20,5), G(31,11), G(47,7)

G(27,9), G(28,10), G(48,8), G(31,11),G(47,7), Q8 f C4, or D8 f C4

G(47,7)√

2 /∈ K , C4, G(18,2), G(20,5), G(31,11), G(46,6)

G(27,9), G(28,10), G(48,8), G(31,11),G(46,6), Q8 f C4, or D8 f C4

G(48,8) G(31,11), G(46,6), G(47,7)

Proof. Let (z,−1) = 1 be an element of Br(K ) guaranteeing the existence of L. For G(16,24), let b ∈ K̇ \ K̇ 2 andchoose a = zb and c = −1. For G(18,2), choose a = z and b = −1. For G(19,4), G(20,5), G(27,9), G(29,14), and G(48,8), choosea = −1, b = z, and x = y = 1. For G(36,28), choose a = z and c = −1, and for G(39,31), choose b = z and c = −1.Realizability of G(31,11), G(46,6), G(47,7), and D8 f C4 then follows from Theorem 3.1.To show realizability of G(28,10) and G(30,13) we consider cases depending on the behavior of 2. If ±2 ∈ K̇ 2, let a = z andb = −1. If ±2 /∈ K̇ 2, then −1 and 2 are quadratically independent in K , and we can let a = 2 and b = −1. (In thiscase we may take z = 2 to yield the desired C4-extension L.)Using Theorem 2.2, part 2, bullet 3, the group G(32,15) is realizable if s(K ) = 2 and there exists a /∈ −K̇ 2 with(a, 2) = 1 ∈ Br(K (i)). This certainly holds if (a, 2) = 1 ∈ Br(K ). If 2 ∈ K̇ 2, this is trivially satisfied. If 2 ∈ −K̇ 2, thenwe can choose a = z, and again the condition is satisfied. If 2 /∈ ±K̇ 2, we can choose a = 2.

Theorem 3.9.Assume s(K ) = 2 and suppose K has a D8-extension M that does not contain

√−1. Then the groups G(17,38), G(33,27),

G(38,30), G(44,43), and G(45,44) are all realizable over K . If in addition |K̇ /K̇ 2| ≥ 16, then G(43,50) is realizable.

Proof. Let (z, w) = 1 be an element of Br(K ) guaranteeing the existence of the D8-extension. For G(17,38), choosea = z, b = w, c = −1, and x = 1. For G(33,27), choose a = z, b = −w, and c = −1. For G(38,30), choose a = −z, b = w,

253


and c = −1. For G(44,43) and G(45,44), choose c = −1, reducing the criterion to (b, 2a) = 1 for both groups. If√

2 ∈ K ,choose a = z and b = w. If

√2 /∈ K , choose a = 2. If |K̇ /K̇ 2| ≥ 16, choosing b = −1, c = z, and d = w yields the

condition for G(43,40).

We summarize the easily stated results for fields of level two in Table 5. As in Table 4, for each group listed in the firstcolumn, the second column lists groups whose realizability implies its realizability, sometimes with additional conditionsthat must be fulfilled for the automatic realizability results to hold. These conditions involve the presence or absence of√

−1 in K or in some D8-extension, M/K . The third column lists groups whose (nontrivial) realizability is guaranteedby the realizability of the given group. We continue to assume that K contains at least as many independent squareclasses as the rank of the group under consideration.

Table 5. Realizability Results Over Fields of Level Two

Group G(m,n) is realizable if G(m,n) implies

G(16,24)√

−1 /∈ LG(17,38)

√−1 /∈ M

G(18,2)√

−1 /∈ L G(31,11), G(46,6), G(47,7)

G(19,4)√

−1 /∈ LG(20,5)

√−1 /∈ L G(31,11), G(46,6), G(47,7)

G(21,12) alwaysG(27,9)

√−1 /∈ L G(31,11), G(46,6), G(47,7)

G(28,10)√

−1 /∈ L G(31,11), G(46,6), G(47,7)

G(29,14)√

−1 /∈ LG(30,13)

√−1 /∈ L

G(31,11)√

−1 /∈ L, G(18,2), G(20,5), G(46,6), G(47,7)

G(27,9), G(28,10), G(46,6),G(47,7), G(48,8), or D8 f C4

G(32,15)√

−1 /∈ LG(33,27)

√−1 /∈ M

G(34,34) alwaysG(35,35) alwaysG(36,28)

√−1 /∈ L

G(38,30)√

−1 /∈ MG(39,31)

√−1 /∈ L

G(43,50)√

−1 /∈ MG(44,43)

√−1 /∈ M

G(45,44)√

−1 /∈ MG(46,6)

√−1 /∈ L, G(18,2), G(20,5), G(31,11), G(47,7)

G(27,9), G(28,10), G(31,11),G(47,7), G(48,8), or D8 f C4

G(47,7)√

−1 /∈ L, G(18,2), G(20,5), G(46,6), G(47,7)

G(27,9), G(28,10), G(31,11),G(46,6), G(48,8), or D8 f C4

G(48,8)√

−1 /∈ L G(31,11), G(46,6), G(47,7)

When s(K ) = 1, the realizability conditions for the groups of order 32 simplify significantly. We begin by creating setsof groups whose resulting realizability conditions are equivalent.

• Let A = {G(20,5), G(21,12)}. The realizability condition for either of these groups is (a, b) = (a, 2) = 1 for quadrati-cally independent a, b in K̇ .

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• Let B = {G(26,42), G(27,9), G(28,10), G(29,14), G(30,13), G(32,15)}. The realizability condition for any and all of these groupsis (a, b) = (a, 2)(b, x) = 1 for some x and quadratically independent a, b in K̇ .

• Let C = {G(33,27), G(34,34), G(35,35), G(36,28), G(37,29), G(38,30), G(39,31), G(40,32), G(41,33)}. The realizability condition forany and all of these groups is (a, c) = (b, c) = 1 for quadratically independent a, b, c in K̇ .

• Let D = {G(44,43), G(45,44)}. The realizability condition for either of these groups is (b, 2a)(c, x) = (b, c) = 1 forsome x and quadratically independent a, b, c in K̇ .

• Let E = {G(16,24), G(18,2), G(31,11), G(46,6), G(47,7), G(48,8), D8}. The realizability condition for any and all of the thesegroups is (a, b) = 1 for quadratically independent a, b in K̇ .

• Let F = {G(42,49), G(43,50)}. The realizability condition for either of these groups is (a, b)(c, d) = 1 for quadraticallyindependent a, b, c, d in K̇ . In particular it is necessary that |K̇ /K̇ 2| ≥ 16 for either of these groups to berealizable.

The majority of these equivalences follow immediately from the results in Table 3 together with the substitution of 1 for−1, since −1 is a square in K̇ , and relabeling of elements a, b, c as needed. We discuss here two groups for which theresults are less obvious. First consider the group G(26,42), with realizability condition (a, c)(b, 2)(c, x) = (b, c) = 1, when−1 ∈ K̇ 2. Relabeling a as c, b as a, and c as b gives (c, b)(a, 2)(b, x) = (a, b) = 1. Letting x ′ = cx yields a conditionidentical to that for G(27,9) when −1 ∈ K̇ 2.Next consider the group G(33,27) with realizability condition (a, bc) = (b, c) = 1, when s(K ) = 1. Since a, b, c are allquadratically independent, so are a, b, bc. Relabeling c as bc, and noting that when s(K ) = 1, (b, b) = 1, yields acondition identical to that for G(34,34) in the case when s(K ) = 1.For convenience, we present these equivalence classes in Table 6.

Table 6. Groups with Equivalent Realizability Conditions when s(K ) = 1

Name Groups

A G(20,5), G(21,12)

B G(26,42), G(27,9), G(28,10), G(29,14), G(30,13), G(32,15)

C G(33,27), G(34,34), G(35,35), G(36,28), G(37,29)

D G(44,43), G(45,44)

E G(16,24), G(18,2), G(31,11), G(46,6), G(47,7), G(48,8), D8

F G(42,49), G(43,50)

Theorem 3.10.Assume s(K ) = 1. Realizability of either of the groups in A or any of the groups D16, Q16, and SD16 (each of order16) implies realizability of all of the groups in B . Realizability of any of the groups in B , C, or D or any of G(49,18),G(50,19), G(51,20), or G(32,15) implies realizability of all of the groups in E . Realizability of any of the groups in E impliesrealizability of G(17,38) and G(19,4).

Proof. Let (a, b) = (a, 2) = 1 be elements of Br(K ) guaranteeing the existence of the groups in A. Letting x = 1gives us the realizability of the groups in B . Using [4, Proposition 2.9], when s(K ) = 1, the realizability condition for eachof D16, Q16, and SD16 simplify to (a, b) = (b, 2)(2x, ab) = 1. Replacing b with ab yields (a, ab) = (ab, 2)(2x, b) = 1.Since (a, a) = 1 and (ab, 2)(2x, b) = (a, 2)(x, b), this gives the realizability condition for the groups in B .For realizability of the groups in E , observe that all of the groups in B , C, D , as well as G(49,18), G(50,19), G(51,20), andG(32,15), have (a, b) = 1 as part of their realizability condition, which is the criterion for realizability of all of the groupsin E .Now let (z, w) = 1 be an element of Br(K ) guaranteeing the realizability of the groups in E . If

√2 ∈ K , G(17,38) and

G(19,4) are seen to be realizable by letting a = z and b = w. If√

2 /∈ K , choose b = 2 to see that G(19,4) is realizable.

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For G(17,38) we consider two cases. First, assuming z, w, and 2 are quadratically independent, let a = z, b= w, andc = 2. If, instead, z, w, and 2 are quadratically dependent, notice that (zw,w) = (z, zw) = 1, and so we may assumewithout loss of generality that w = 2. Letting c be any element of K̇ quadratically independent of w and z, we choosea = zc and b = w = 2 and the criterion for G(17,38) becomes (2c, 2)(c, 2) = (2, 2) which is trivial, as desired.

Theorem 3.11.Assume s(K ) = 1. If

√2 /∈ K , then G(19,4) is realizable over K . If

√2 ∈ K , then the realizability of any of the groups in

A, B , or E or either of G(17,38) or G(19,4) implies the realizability of all of the others, and the realizability of any of thegroups in C implies the realizability of all of the groups in D .

Proof. If√

2 /∈ K , choosing b = 2 gives the realizability of G(19,4). Now assume that√

2 ∈ K . The realizabilitycriteria for all of the groups in A, B , and E , as well as for G(17,38) and G(19,4), collapse to (a, b) = 1. Switching thelabeling of b and c and choosing x = 1 proves the final statement.

Tables 7 and 8 list realizability criteria for each of the non-direct product groups of order 32, under the condition thats(K ) = 1. For each group listed in the first column, the second column lists groups with equivalent realizability. Thethird column lists groups whose realizability implies the realizability of the given group. The fourth column lists groupswhose (nontrivial) realizability is guaranteed by the realizability of the given group. As noted, some of these realizabilityconditions depend on the existence of

√2 in K . As in Tables 4 and 5, it is assumed that K contains at least as many

independent square classes as the rank of the group G under consideration. Groups for which no easily stated automaticrealizability results have been obtained have been excluded from the table.

4. Conclusion

In this work, we have presented necessary and sufficient conditions for the realizability of each group of order 32 asa Galois group over a field K in terms of the triviality of specific elements in the Brauer group of K and of certainquadratic extensions of K . This comprehensive list of criteria provides a unified presentation of results obtained bymany different researchers, and heretofore scattered throughout the literature. We also have stated and, using the aboveresults, proved a wide range of automatic realizability results, wherein the realizability of one group as a Galois groupover K (with perhaps some minimal additional conditions) implies the realizability of another group that is not simplya quotient group of the first.

Acknowledgements

The first author acknowledges the support of the NSF-AWM Michler Collaborative Research Grant.

References

[1] Carlson J., The mod 2 cohomology of 2-groups, Tables of 2-groups: 〈http://www.math.uga.edu/̃ lvalero/cohointro.html〉[2] Gao W., Leep D.B., Minác J., Smith T.L., Galois groups over nonrigid fields, In: Valuation Theory and its Applications,

Vol. II, Fields Institute Communications Series, 33, American Mathematical Society, 2003, 61–77[3] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.9, 2006, 〈http://www.gap-system.org〉[4] Grundman H.G., Smith T.L., Automatic realizability of Galois groups of order 16, Proc. AMS, 1996, 124, 2631–2640[5] Grundman H.G., Smith T.L., Galois realizability of a central C4-extension of D8, J. Alg., 2009, 322, 3492–3498[6] Grundman H.G., Smith T.L., Swallow J.R., Groups of order 16 as Galois groups, Expo. Math., 1995, 13, 289–319[7] Grundman H.G., Stewart G., Galois realizability of non-split group extensions of C2 by (C2)r × (C4)s × (D4)t , J.

Algebra, 2004, 272, 425–434[8] Hall M.Jr., Senior J.K., The Groups of Order 2n (n ≤ 6), Macmillian, New York, 1964

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Table 7. Realizability Results Over Fields of Level One, Part I

Group Equivalent Groups G(m,n) is realizable if G(m,n) implies

G(16,24) E ; if√

2 ∈ K , also D16, Q16, SD16, A, B , C, DA, B , G(17,38), G(19,4) G(49,18), G(50,19), or G(51,20)

G(17,38) if√

2 ∈ K , D16, Q16, SD16, A, B , C, D ,A, B , E , G(19,4) E , G(49,18), G(50,19), or G(51,20)

G(18,2) E ; if√

2 ∈ K , also D16, Q16, SD16, A, B , C, DA, B , G(17,38), G(19,4) G(49,18), G(50,19), or G(51,20)

G(19,4) if√

2 ∈ K , D16, Q16, SD16, A, B , C, D ,A, B , E , G(17,38) E , G(49,18), G(50,19), or G(51,20)

G(20,5) A; if√

2 ∈ K , also both√

2 ∈ K and C EB , E , G(17,38), G(19,4)

G(21,12) A; if√

2 ∈ K , also both√

2 ∈ K and C EB , E , G(17,38), G(19,4)

G(26,42) B ; if√

2 ∈ K , also D16, Q16, SD16, A, or EA, E , G(17,38), G(19,4) both

√2 ∈ K and C

G(27,9) B ; if√


√2 ∈ K and C

G(28,10) B ; if√


√2 ∈ K and C

G(29,14) B ; if√


√2 ∈ K and C

G(30,13) B ; if√


√2 ∈ K and C

G(31,11) E ; if√

2 ∈ K, also D16, Q16, SD16, A, B , C, DA, B , G(17,38), G(19,4) G(49,18), G(50,19), or G(51,20)

G(32,15) B ; if√


√2 ∈ K and C

G(33,27) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(34,34) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

[9] Ishkhanov V.V., Lur’e B.B., Faddeev D.K., The Embedding Problem in Galois Theory, Translations of MathematicalMonographs, 165, American Mathematical Society, Providence, R.I., 1997

[10] Jensen C.U., On the representation of a group as a Galois group over an arbitrary field, In: De Koninck J.-M.,Levesque C. (Eds.), Theorie des Nombres Number Theory, Walter de Gruyter, 1989, 441–458

[11] Kuyk W., Lenstra H.W., Abelian extensions of arbitrary fields, Math. Ann., 1975, 216, 99–104[12] Ledet A., On 2-groups as Galois groups, Canad. J. Math., 1995, 47, 1253–1273[13] Ledet A., Embedding problems with cyclic kernel of order 4, Israel J. Math., 1998, 106, 109–131[14] Michailov I., Embedding obstructions for the cyclic and modular 2-groups, Math. Balkanica (N.S.), 2007, 21, 31–50[15] Michailov I., Groups of order 32 as Galois groups, Serdica Math. J., 2007, 33, 1–34[16] Smith T.L., Extra-special 2-groups of order 32 as Galois groups, Canad. J. Math., 1994, 46, 886–896[17] Swallow J., Thiem N., Quadratic corestriction, C2-embedding problems, and explicit construction, Comm. Algebra,

2002, 30, 3227–3258[18] Witt E., Konstruktion von Galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung pf , J.

Reine Angew. Math., 1936, 174, 237–245 (in German)

257


Table 8. Realizability Results Over Fields of Level One, Part II

Group Equivalent Groups G(m,n) is realizable if G(m,n) implies

G(35,35) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(36,28) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(37,29) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(38,30) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(39,31) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(40,32) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(41,33) C E ; if√

2 ∈ K, also A,B , D , G(17,38), G(19,4)

G(42,49) FG(43,50) FG(44,43) D both

√2 ∈ K and C E ; if

√2 ∈ K, also

A, B , G(17,38), G(19,4)

G(45,44) D both√

2 ∈ K and C E ; if√

2 ∈ K, alsoA, B , G(17,38), G(19,4)

G(46,6) E ; if√


G(47,7) E ; if√


G(48,8) E ; if√


G(49,18) E ; if√

2 ∈ K , alsoA, B , G(17,38), G(19,4)

G(50,19) E ; if√

2 ∈ K , alsoA, B , G(17,38), G(19,4)

G(51,20) E ; if√

2 ∈ K , alsoA, B , G(17,38), G(19,4)

258


Table 9. Generators and Relations for Nonabelian Non-Direct Products, I

Group Exp Rk Relations Z (G) G/Z (G)

G(16,24) 4 3 x4 = y2 = z4 = 1; [y, x] = z2 〈z, x2〉 C2 × C2

G(17,38) 8 3 x8 = y2 = z2 = 1; [z, y] = x4 〈x〉 C2 × C2

G(18,2) 4 2 x4 = y4 = [y, x]2 = 1 〈x2, y2, [y, x]〉 C2 × C2

G(19,4) 8 2 x8 = y4 = 1; [y, x] = x4 〈x2, y2〉 C2 × C2

G(20,5) 8 2 x8 = y2 = [y, x]2 = 1 〈x2, [y, x]〉 C2 × C2

G(21,12) 8 2 x8 = y4 = 1; [y, x] = y2 〈x2, y2〉 C2 × C2

G(22,17) 16 2 x16 = 1, y2 = 1, [y, x] = x8 〈x2〉 C2 × C2

G(26,42) 8 3 x2 = y2 = z4 = 1; [y, x] = w; 〈z〉 D8

w2 = z2 = [w, x] = [w, y]G(27,9) 8 2 x4 = y2 = z4 = 1; 〈x2, z2〉 D8

[y, x] = z; [z, x] = [z, y] = z2

G(28,10) 8 2 x4 = y4 = 1; [y, x] = z 〈x2, y2〉 D8

y2 = z2 = [z, x] = [z, y]G(29,14) 8 2 x4 = y8 = 1; [y, x] = y6 〈x2, y4〉 D8

G(30,13) 8 2 x4 = y8 = 1; [y, x] = y2 〈x2, y4〉 D8

G(31,11) 8 2 x8 = y2 = 1; [y, x] = z; 〈x2〉 D8

x4 = [z, x] = [z, y] = z2

G(32,15) 8 2 x8 = 1; x2 = y2; [y, x] = z; 〈x2〉 D8

x4 = [z, x] = [z, y] = z2

G(33,27) 4 3 x2 = y2 = z2 = 1; 〈[y, x], [z, x]〉 (C2)3

[y, x]2 = [z, x]2 = [z, y] = 1G(34,34) 4 3 x2 = y4 = z4 = [y, z] = 1; 〈y2, z2〉 (C2)3

[y, x] = y2; [z, x] = z2

G(35,35) 4 3 x4 = y4 = z4 = [z, y] = 1; 〈y2, z2〉 (C2)3

x2 = y2 = [y, x]; [z, x] = z2

G(36,28) 4 3 x2 = y4 = z2 = [z, y] = 1; 〈y2, [z, x]〉 (C2)3

[y, x] = y2; [z, x]2 = 1G(37,29) 4 3 x4 = z2 = [z, y] = [z, x]2 = 1; 〈x2, [z, x]〉 (C2)3

x2 = y2 = [y, x]G(38,30) 4 3 x2 = y2 = z4 = [z, y] = 1; 〈z2, [z, x]〉 (C2)3

[y, x] = z2; [z, x]2 = 1

259


Table 10. Generators and Relations for Nonabelian Non-Direct Products, II

Group Exp Rk Relations Z (G) G/Z (G)

G(39,31) 4 3 x2 = y4 = [y, x]2 = 1; 〈y2, [y, x]〉 (C2)3

y2 = z2 = [z, x] = [z, y]G(40,32) 4 3 x4 = y4 = z4 = [y, x] = 1; 〈x2, y2〉 (C2)3

[z, x] = z2; [z, y] = y2; x2 = y2z2

G(41,33) 4 3 x2 = y4 = z4 = [z, y] = 1; 〈y2, z2〉 (C2)3

[y, x] = z2; [z, x] = y2z2

G(42,49) 4 4 x2 = y4 = z2 = w2 = 1; 〈[y, x]〉 (C2)4

[y, x] = [z, w] = y2 = w2;[z, x] = [z, y] = [w, x] = [w, y] = 1

G(43,50) 4 4 x2 = y4 = z4 = w4 = 1; 〈y2〉 (C2)4

[z, x] = [w, x] = [z, y] = [w, y] = 1;y2 = z2 = w2 = [y, x] = [z, w]

G(44,43) 8 3 x2 = y2 = z2 = w4 = 1; 〈w2〉 D8 × C2

[z, x] = [w, x] = 1; [y, x] = w;[y, z] = [w, z] = [w, y] = w2

G(45,44) 8 3 x4 = y2 = z2 = 1; 〈x2〉 D8 × C2

[z, x] = w; [y, x] = [w, y] = 1;[z, y] = [w, x] = [w, z] = x2 = w2

G(46,6) 8 2 x4 = y2 = z2 = 1; [z, x] = [x2, y]; 〈[z, x]〉 D8 f C4

[y, x] = z; [z, x]2 = [z, y] = [x2, z] = 1G(47,7) 8 2 x8 = y2 = z2 = 1; [y, x] = z; 〈x4〉 D8 × C2

[z, x] = [x2, y] = x4; [z, y] = [x2, z] = 1G(48,8) 8 2 x8 = y4 = z2 = 1; x4 = y2; [y, x] = z 〈y2〉 D8 × C2

[z, x] = [x2, y] = y2; [z, y] = [x2, z] = 1G(49,18) 16 2 x16 = y2 = 1; [y, x] = x2 〈x8〉 D16

G(50,19) 16 2 x16 = y2 = 1; [y, x] = x10 〈x8〉 D16

G(51,20) 16 2 x16 = 1; x8 = y2; [y, x] = x2 〈x8〉 D16

260

Realizability and automatic realizability of Galois groups of order 32

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