An Errata for: Torsion subgroups of rational elliptic curves over the compositum of all D_4 extensions of the rational numbers
aa r X i v : . [ m a t h . N T ] F e b AN ERRATA FOR: TORSION SUBGROUPS OF RATIONAL ELLIPTICCURVES OVER THE COMPOSITUM OF ALL D EXTENSIONS OF THERATIONAL NUMBERS
HARRIS B. DANIELS
Abstract.
In [2], the author claims that the fields Q ( D ∞ ) defined in the paper and the compositumof all D extensions of Q coincide. The proof of this claim depends on a misreading of a celebratedresult by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification oftorsion structures of E defined over Q when base changed to the compositum of all D extensionsof Q main results of [2]. All the main results in [2] are still correct except that we are no longer ableto prove that these two fields are equal. Introduction
The issue that this paper addresses arises at the end of [2, Section 1] where the author asserts thatthe compositum of all D extensions of Q , called F in that paper and Q D here, and the compositumof all generalized D extensions of Q , see [2, Definition 1.10], denoted Q ( D ∞ ), are the same. Theauthor says that this follows from a simple argument as well as a celebrated theorem of Shafarevich.The argument is made that since D is nilpotent the kernels of all the embedding problems mustalso be nilpotent and hence are solvable. The problem is that there is another condition that mustbe met in order to apply said theorem. For the reader’s convenience we include a statement of thetheorem by Shafarevich below. Theorem 1.1. [6, 10, 11]
Let
L/K be an extension of number fields with Galois group S , let U be a nilpotent group with S -action, and let G be a semi-direct product U ⋊ S . Then the embeddingproblem for L/K and → U → G → S → has a solution. As Theorem 1.1 indicates, in order to use this theorem it is not sufficient for the embeddingproblem to have a nilpotent kernel, it must also be split. While D itself fits into a short exactsequence that is split, it is not the case that all the relevant embedding problems can be made tosplit. Example 1.2.
Consider K = Q ( √ d ). It is a classical result that K can only be embedded into adegree 4 cyclic extension of Q if d can be written as the sum of two rational squares. In order words,the embedding problem given by 1 → Z / Z → Z / Z → Z / Z → Mathematics Subject Classification.
Primary: 11G05, Secondary: 11R21, 12F10, 14H52.
Key words and phrases.
Elliptic Curve, Torsion Points, Galois Theory.
In contrast, the problem of embedding a single quadratic extension of Q into a D extension of Q is always possible since the relevant embedding problem1 → Z / Z → D ≃ Z / Z ⋊ Z / Z → Z / Z → L/ Q with Galois group D in fact contains 3 different quadratic extensions of Q . Using the Galois correspondence, we can seethat the embedding problem above corresponds to embedding Q ( √ d ) into L in the only way thatmakes the extension L/ Q ( √ d ) have Galois group Z / Z .For the rest of the article we will let Q D be the compositum of all D extensions of Q , Q ( D ∞ ) bethe compositum of all generalized D extensions of Q , and we aim to prove the following theorem: Theorem 1.3.
Let E/ Q be an elliptic curve. Then E ( Q D ) tors = E ( Q ( D ∞ )) tors . Once Theorem 1.3 is proven all of the results in [2] are completely justified except the assertionthat Q D = Q ( D ∞ ). Since this assertion was not a central part of the paper we don’t attempt toprove it here, but we do discuss what is left to show in Section 4.For the ease of the reader, in what follows any groups defined by a small group label will have alink to the corresponding LMFDB page [7] if it exists and if there is a more commonplace name ordescription, we use that in place of the small group label.1.1. Acknowledgements.
We would like to thank Maarten Derickx for pointing this mistake outand for helpful conversations throughout the preparation of this document. We would also like tothank Jackson Morrow and the anonymous referee for helpful comments on an initial draft of thispaper. 2.
Galois Embedding Problems and the Brauer Group of a Field
We start this section by introducing Galois embedding problems and exploring a particular prob-lem that illustrates the need for this paper. For the remainder of this section unless otherwise statedwe will let
L/K be a Galois extension with Galois group Q and1 → N → G → Q → M/K with L ⊆ M and Gal( M/K ) ≃ G . If such an M exists, we say that M is a solution to the embeddingproblem given by ( L/K, G, N ).One of the main tools that is used to study Galois embedding problems is the Brauer group of afield.
Definition 2.1.
Let K be a field. The Brauer group of K , denoted Br( K ) , is the abelian groupwhose elements are Morita equivalence classes of central simple algebras over K under the binaryoperation of tensor products. For a, b ∈ K we let ( a, b ) K (or just ( a, b ) when it is clear from context) represent the class ofcentral simple algebras equivalent to the quaternion algebra K ( i, j ) / h i = a, j = b, ij = − ji i insideBr( K ). These algebras have order 2 in Br( K ) and are deeply connected to the embedding problemsthat we are interested in. ORSION OVER Q ( D ∞ ) – ERRATA 3 The main result that we will use to prove Theorem 1.3 is known as the embedding criterion andit gives criterion for the existence of a solution to the embedding problem of the form( F ( √ a , . . . , √ a r ) /F, G, Z / Z ) , where F is a field, the a i ’s are independent modulo ( F × ) , and G is a non-split central extension of( Z / Z ) r . Theorem 2.2. (Embedding Criterion, [4, Section 7])
Let F be a field and K = F ( √ a , . . . , √ a r ) were the a i ’s are in F × and independent modulo ( F × ) . Let Q = Gal( K/F ) , and consider a non-split central extension G of Z / Z by Q . Let σ , . . . , σ r generate Q , where σ i ( √ a j ) = ( − δ ( i,j ) √ a j and let ˜ σ , . . . , ˜ σ r be any set of preimages of σ , . . . , σ r in G . Define c ij for i ≤ j by c ij = 1 if andonly if [˜ σ i , ˜ σ j ] = 1 for i < j and c ii = 1 if and only if (˜ σ i ) = 1 . There exists a Galois Extension L/F , L ⊇ K , such that Gal(
L/F ) ≃ G and the surjection G → Q is the natural surjection of Galoisgroups, if and only if Y i ≤ j ( a i , a j ) c ij = 1 ∈ Br( F ) . Using Theorem 2.2, we get the following result concerning polyquadratic extensions of Q . Corollary 2.3. [5, 12]
Let a, b, c, d ∈ Z be nonsquare and independent mod Z .(1) The field Q ( √ a ) can be embedded into a Z / Z extension of Q if and only if ( a, a ) = ( a, −
1) = 1 ∈ Br( Q ) . (2) The field Q ( √ a, √ b ) can be embedded into a D extension of Q if and only if ( a, b ) = 1 ∈ Br( Q ) . (3) The field Q ( √ a, √ b ) can be embedded into a Q extension of Q if and only if ( a, b )( ab, −
1) = 1 ∈ Br( Q ) . (4) The field Q ( √ a, √ b, √ c ) can be embedded into a SmallGroup(16 , extension of Q if andonly if ( a, ab )( c, −
1) = 1 ∈ Br( Q ) . (5) The field Q ( √ a, √ b, √ c, √ d ) can be embedding into a SmallGroup(32 , extension of Q ifand only if ( a, b )( c, d ) = 1 ∈ Br( Q ) . We will use the information in Corollary 2.3 in order to prove that if K/ Q has Galois group G which is isomorphic to one of the 5 groups in Corollary 2.3, then K must be in Q D . Remark 2.4.
While it may not seem like it at first glance, everything in Corollary 2.3 is completelyexplicit. For example, given a field K = Q ( √ a, √ b ) such that ( a, b ) = 1 ∈ Br( Q ), every D extensionof Q that contains K can be given concretely. Since ( a, b ) = 1 ∈ Br( Q ) there exist α, β ∈ Q suchthat b = α − aβ . Then for any r ∈ Q × , the extension Q ( p r ( α + β √ a ) , √ β ) / Q is a D extensionthat contains K and every such extension arises in this way. HARRIS B. DANIELS Proof of the main theorem
Before outlining the proof of Theorem 1.3, we give a definition and lemma.
Definition 3.1.
Given a group G , we say that a field F/ Q is G -complete if for every Galoisextension K/ Q such that Gal( K/ Q ) ≃ G it follows that K ⊆ F . The field F is said to be G -incomplete if there exists at least one Galois extension K/ Q with Gal( K/ Q ) ≃ G and K F . Lemma 3.2.
Let F be an extension of Q and let K/ Q be a finite Galois extension with Galois group G . If there exist normal subgroups N , . . . , N r ✁ G such that(1) for all ≤ i ≤ r the field F is G/N i -complete, and(2) r \ i =1 N i = { e } ,then K ⊆ F .Proof. For each 1 ≤ i ≤ r let K i be the subfield of K with Galois group N i guaranteed to existsby the fundamental theorem of Galois theory. Each of these fields is in F by condition (1) andcondition (2) ensures that K = K K . . . K r . Since each K i is in F we have that K = K K . . . K r is contained in F . (cid:3) Remark 3.3.
Clearly, if K/ Q is a finite extension that is not Galois with Galois closure K Gal suchthat K Gal / Q satisfies conditions of Lemma 3.2 with F = Q D , then K ⊆ K Gal ⊆ Q D .With Lemma 3.2 in hand, we start to see how one might prove Theorem 1.3. The idea is to firstshow that Q D is G -complete for a few carefully chosen groups G and then use these together withLemma 3.2 to prove that Q ( E ( Q ( D ∞ ))[ p ∞ ]) ⊆ Q D . That is, we will show that for every elliptic curve E/ Q and prime p , the field of definition ofthe p -primary component of E ( Q ( D ∞ )) is contained in Q D . This combined with the fact that Q D ⊆ Q ( D ∞ ) would give us that E ( Q D )[ p ∞ ] = E ( Q ( D ∞ ))[ p ∞ ]. From there, Theorem 1.3 wouldfollow from the classification of finite abelian groups.We start the process by proving that Q D is G -complete for some small groups. Proposition 3.4. If G is a group isomorphic to Z / Z , Z / Z , or D then Q D is G -complete.Proof. Let K/ Q be a number field with Gal( K/ Q ) ≃ G .(1) Suppose G is isomorphic to Z / Z . In this case K = Q ( √ a ) for some squarefree a in Z andwe saw from Theorem 1.1 that any quadratic extension of Q can be embedded into a D extension. Thus K ⊆ Q D and Q D is Z / Z -complete.(2) Suppose G is isomorphic to D . Clearly Q D is D -complete.(3) Suppose G is isomorphic to Z / Z . This means that K has a unique subfield of the form Q ( √ a ). This unique quadratic field can be embedded into a D extension L such that L/ Q ( √ a ) is not a cyclic degree 4 extension and K is not a subfield of L . We now considerthe Galois extension given by the compositum of L and K . Since the intersection of L and K is exactly Q ( √ a ), we have that Gal( LK/ Q ) is isomorphic to a subgroup of index 2 inside D × C , call it G . We also know that if π and π are the projection maps out of D × C ,then π ( G ) = D and π ( G ) = C . Since L ∩ K = Q ( √ a ), G contains a normal subgroup N such that [ D : π ( N )] = [ C : π ( N )] = 2 (so the fixed field that is in both K and L is ORSION OVER Q ( D ∞ ) – ERRATA 5 a quadratic) and π ( N ) = h f i (since Q ( √ a ) / Q is not cyclic). If we let N and N be the 2,noncyclic subgroups of D of index 2, we can rephrase these conditions to say that G has theproperty that either G ∩ π − ( N ) = G ∩ π − ( h c i ) or G ∩ π − ( N ) = G ∩ π − ( h c i ) and thecommon group is an index 2 subgroup of G . Searching the index 2 subgroups of D × C ,we see that there are two groups up to conjugation with this property and they are bothisomorphic to SmallGroup(16 , LK/ Q ) is isomorphic to SmallGroup(16 , LK/ Q ) = H we see that H has two distinct normal subgroups N and N such that N ∩ N = { e } and H/N ≃ H/N ≃ D . Thus by Lemma 3.2 and part (2) of thisproposition, K ⊆ KL ⊆ Q D and Q D is Z / Z -complete (cid:3) The following lemmas give a list of small groups for which Q D is not necessarily G -complete, butwith some added assumptions on K/ Q , we can show that K ⊆ Q D . Lemma 3.5.
Let G be isomorphic to SmallGroup(8,4) and let F = Q ( √ a, √ b ) such that ( a, b ) = ( ab, −
1) = 1 ∈ Br( Q ) . If K/ Q is a solution to the embedding problem ( F/ Q , G, Z / Z ) , then K ⊆ Q D .Proof. From Corollary 2.3 and the added assumption that ( a, b ) = 1 ∈ Br( Q ), we know that there isa Galois extension L/ Q such that Q ( √ a, √ b ) ⊆ L and Gal( L/ Q ) ≃ D . Considering the compositumof K and L we see that Gal( LK/ Q ) is isomorphic to the uniqe group of order 16 with a quotientisomorphic to D Q , namely H ≃ Smallgroup(16 , H has 3 normal subgroup N , N , and N such that N ∩ N ∩ N = { e } and H/N ≃ D and H/N ≃ H/N ≃ Z / Z . Therefore, by Lemma 3.2 and Proposition 3.4 , we have K ⊆ LK ⊆ Q D . (cid:3) Remark 3.6.
Here we point out that the previous proof, we didn’t actually need all 3 of the normalsubgroups. In fact, N ∩ N = N ∩ N = { e } and so one in fact only needs two fields tow generate L . We included all 3 in the proof because this is the way the search was conducted, by computing all the relevant normal subgroups and computing their intersection. The code used to verify all ofthese computations can be found at [3]. Lemma 3.7.
Let G be isomorphic to SmallGroup(16,13) and let F = Q ( √ a, √ b, √ c ) such that ( a, ab ) = ( c, −
1) = 1 ∈ Br( Q ) . If K/ Q is a solution to the embedding problem ( F/ Q , G, Z / Z ) , then K ⊆ Q D .Proof. From Corollary 2.3 and the assumption that ( c, −
1) = 1 ∈ Br( Q ) the field Q ( √ c ) can beembedded into a degree 4 cyclic extension L/ Q . Again considering the field LK we compute thatGal( LK/ Q ) ≃ D × Z / Z ≃ SmallGroup(32 ,
25) and thus we have K ⊆ LK ⊆ Q D . (cid:3) Lemma 3.8.
Let G be isomorphic to SmallGroup(32,49) and let F = Q ( √ a, √ b, √ c, √ d ) such that ( a, b ) = ( c, d ) = 1 ∈ Br( Q ) . If K/ Q is a solution to the embedding problem ( F/ Q , G, Z / Z ) , then K ⊆ Q D .Proof. From Corollary 2.3 and the added the assumption that ( a, b ) = 1 ∈ Br( Q ) we have that Q ( √ a, √ b ) can be embedded into a D extension, we pick one of the possible solutions to thisembedding problem and call it L . Again we consider the extension LK/ Q and compute thatGal( LK/ Q ) = H ≃ SmallGroup(64 , H has 3 normal subgroups N , N , and HARRIS B. DANIELS N , such that N ∩ N ∩ N = { e } , H/N ≃ H/N ≃ D , and H/N ≃ SmallGroup(16 , K ⊆ LK ⊆ Q D . (cid:3) Before we can prove Theorem 1.3 we will need to prove that Q D is G -complete for some groupsof size 64. In order to do this we will first need a group theory lemma and a proposition aboutGalois embedding problems with decomposable kernel. Lemma 3.9. [8, Lemma 4.1]
Let G be a group and let N , and N be normal subgroups of G suchthat N ∩ N = { e } . Then we have that G is isomorphic to the pullback ( G/N ) f ( G/N ) and thefollowing diagram commutes: (cid:15) (cid:15) (cid:15) (cid:15) N (cid:15) (cid:15) N (cid:15) (cid:15) / / N / / G / / (cid:15) (cid:15) G/N / / (cid:15) (cid:15) / / N / / G/N / / (cid:15) (cid:15) G/N N / / (cid:15) (cid:15)
11 1
Here the maps from a group to a quotient of that group are the natural maps.
Using this lemma, one can prove the following proposition.
Proposition 3.10. [8, Theorem 4.1]
Let G , N and N be group as in Proposition 3.9. Let K/ Q be a Galois extension with Gal( K/ Q ) ≃ G/N N . Then the embedding problem ( K/ Q , G, N × N ) is solvable if and only if ( K/ Q , G/N , N ) and ( K/ Q , G/N , N ) are solvable. Remark 3.11.
Proposition 3.10 can be stated more generally for an arbitrary base-field, but weonly state it with Q as the base field so for the sake of simplicity. For more details, the reader isencouraged to see [8].We are now ready to show that F is G -complete with respect to some larger groups. Proposition 3.12.
Let G be isomorphic to SmallGroup(64,206), SmallGroup(64,215), or Small-Group(64,216) . Then Q D is G -complete.Proof. Let K/ Q be a Galois extension with Galois group G , one of the three groups listed above.Then, in all 3 cases [ G, G ] ≃ ( Z / Z ) and G/ [ G, G ] ≃ ( Z / Z ) . Therefore, we can write [
G, G ] = N × N where G , N , and N satisfy the conditions of Lemma 3.9. Choosing N and N correctlywe get that in all cases G/N ≃ D × ( Z / Z ) and G/N ≃ ( SmallGroup(32 , G ≃ SmallGroup(64 , , G ≃ SmallGroup(64 , , . ORSION OVER Q ( D ∞ ) – ERRATA 7 The group SmallGroup(32,48) is isomorphic to SmallGroup(16 , × Z / Z . Using Proposition 3.10,we get that the field Q ( √ a, √ b, √ c, √ d ) can be embedded into a field with Galois group G if andonly if ( ( a, b ) = ( a, b )( c, −
1) = 1 ∈ Br( Q ) G ≃ SmallGroup(64 , a, b ) = ( a, b )( c, d ) = 1 ∈ Br( Q ) G ≃ SmallGroup(64 , , . Rewriting these conditions we get ( ( a, b ) = ( c, −
1) = 1 ∈ Br( Q ) G ≃ SmallGroup(64 , a, b ) = ( c, d ) = 1 ∈ Br( Q ) G ≃ SmallGroup(64 , , . Thus, any Galois extension K/ Q with Gal( K/ Q ) ≃ G is the compositum of two fields of degree32 that are in Q D by Lemmas 3.5, 3.7, and 3.8. Therefore it must be that K ⊆ Q D and Q D is G -complete for all 3 groups. (cid:3) With this we are now ready to prove Theorem 1.3 but before we get started we make a remarkthat will guide our strategy.
Remark 3.13.
Examining the proofs of Propositions 3.4, and 3.12 as well as Lemmas 3.5, 3.7, and3.8, we see that at each step it was useful to think of the field in question as living in a larger butstill finite extension of Q . For example in the proof of Proposition 3.4 while proving that Q D is Z / Z -complete, we start with a generic Z / Z extension of Q , called K/ Q in the proof. The resultfollows from considering K inside of the field LK , where L/ Q is a D extension and [ L ∩ K : Q ] = 2 . Motivated by this, when we are trying to compute the p -primary component of E ( Q D ) and showthat it is Z /p n Z ⊕ Z /p m Z with m ≤ n using this method, it is useful to not just consider the field ofdefinition of the p n -th torsion but actually the field of definition of the p n +1 torsion. On the groupsside of things, this just means that we will lift groups of level p n to level p n +1 before doing any ofthe computations. The idea is that viewing Q ( E [ p n ]) inside Q ( E [ p n +1 ]), allows us to see that therelevant fields are generated by fields that are in Q D . Proof of Theorem 1.3.
Let E/ Q be an elliptic curve and p a prime. We denote the p -primarycomponent of E ( Q ( D ∞ )) by E ( Q ( D ∞ ))[ p ∞ ] and its field of definition by K p = Q ( E ( Q ( D ∞ )[ p ∞ ])) . The overall strategy of the proof is to show that K p ⊆ Q D . This combined with the fact that Q D ⊆ Q ( D ∞ ) would yield that E ( Q ( D ∞ ))[ p ∞ ] = E ( Q D )[ p ∞ ] and from here the result followsfrom the classification of finite abelian groups. We proceed by breaking the argument down intocases based on the size and parity of p .Suppose p ≥ K p / Q is anabelian extension. Since Gal( K p / Q ) has exponent dividing 4 and is abelian, it must be isomorphicto ( Z / Z ) s × ( Z / Z ) s for some nonnegative integers s and s , and since Q D is Z / Z - and Z / Z -complete K p ⊆ Q D and E ( Q ( D ∞ ))[ p ∞ ] = E ( Q D )[ p ∞ ].Next, suppose p = 3. From [2, Section 5.4], we see that there are 3 different cases that needto be checked corresponding to the possible isomorphism class of E ( Q ( D ∞ ))[3 ∞ ]. For two of themGal( K p / Q ) ≃ Z / Z and in this case clearly K ⊆ Q D . The last cases corresponds to the casewhen Gal( K / Q ) is isomorphic to a subgroup of the normalizer of the split Cartan subgroup ofGL ( Z / Z ). It turns out that the normalizer of the non-split Cartan subgroup of GL ( Z / Z ) isisomorphic to D and again we have K ⊆ Q D . HARRIS B. DANIELS
The last outstanding case is when p = 2. To settle this case, let S be the set containing ofisomorphism classes of groups with small groups label (2,1),(4,1),(8,3),(64,206), (64,215), or (64,216).When a we say a group is in S , we mean that it is isomorphic to a group in S .Next for each of the relevant groups from [9] and as discussed in Remark 3.13, consider them attwice the level necessary to verify [2, Table 2]. For each of these groups G , we let S G = { N : N ⊳ G and
G/N ∈ S} and N G = \ N ∈ S G N. Now, if E/ Q is comes from a point on the modular curve corresponding to one of the groups G in [2, Table 2], then there is a subfield of Q ( E [2 n ]) call it L G corresponding to the fixed fieldof N G and with Galois groups G/N G . The field L G is contained in Q D since Q D is G completefor all G ∈ S by Propositions 3.4 and 3.12. Further we know that E ( L G )[2 ∞ ] is isomorphic to thesubgroup of ( Z / n Z ) that is fixed by N G where 2 n is the level of G . So if we can confirm that E ( L G )[2 ∞ ] ≃ E ( Q ( D ∞ ))[2 ∞ ], then we would have E ( Q D )[2 ∞ ] ≃ E ( Q ( D ∞ ))[2 ∞ ] since E ( L G ) ⊆ E ( Q D ) ⊆ E ( Q ( D ∞ )). All that is left to do is show that for each G of interest the fixed groupof N G is isomorphic to the subgroup given in [2, Table 2]. Thus E ( D ∞ )[2 ∞ ] = E ( K G )[2 ∞ ] ⊆ E ( Q D )[2 ∞ ] ⊆ E ( D ∞ )[2 ∞ ] implying that E ( Q D )[2 ∞ ] = E ( Q ∞ D )[2 ∞ ]. Thus completing the casewhen p = 2 as well as the proof of Theorem 1.3. The code confirming this can be found in [3]. (cid:3) Remaining Open Questions
The question about the equality of the fields Q ( D ∞ ) and Q D still remains open, but there aresome smaller questions one could ask that could potentially shed light on the relationship betweenthese two fields. Question 4.1. Is Q D a Q -complete field?We saw that under the added assumption that each individual term coming from the embeddingcriterion is trivial we can show that a Q extension of Q is in Q D , but the question remains in thecase when ( a, b ) = ( ab, − = 1 ∈ Br( Q ). More concretely, consider the field Q ( α ) / Q where α is aroot of f ( x ) = x + 84 x + 1260 x + 5292 x + 441 . In this case, we have that Gal( K/ Q ) ≃ Q and Q ( √ , √ ⊂ K and (3 ,
14) = (42 , − = 1 ∈ Br( Q ).Is K ⊆ Q D ? We suspect that if one could find the answer to this particular questions, they couldalso likely settle the relationship between Q D and Q ( D ∞ ). References [1] W. Bosma, J. Cannon, and C. Playoust,
The Magma algebra system. I. The user language , J. Symolic Comput., (1997), 235–265.[2] H. B. Daniels, Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rationalnumbers,
J. Algebra 509 (2018), 535–565.[3] H. B. Daniels, Magma scripts related to
An errata for: Torsion subgroups of rational elliptic curves over thecompositum of all D extensions of the rational numbers , available at http://hdaniels.people.amherst.edu [4] A. Fr¨ohlich, Orthogonal representations of Galois groups, Stiefel-Whitney classes and Hasse-Witt invariants.
J.Reine Angew. Math. 360 (1985), 84–123.[5] H. G. Grundman and T. L. Smith,
Groups of order 16 as Galois groups , Proc. Amer. Math. Soc. 124 (1996),2631-2640
ORSION OVER Q ( D ∞ ) – ERRATA 9 [6] V. V. Ishkhanov, On the semi-direct embedding problem with nilpotent kernel , Izv. Akad. Nauk SSSR 40 (1976),3-25.[7] LMFDB Collaboration,
The L-functions and modular forms database , available at .[8] I. Michailov
Groups of order 32 as Galois groups,
Serdica Math. J. 33 (2007), no. 1, 1–34.[9] J. Rouse, D. Zureick-Brown,
Elliptic curves over Q and -adic images of Galois , Research in Number Theory (2015), 34 pages.[10] J.-P. Serre, Topics in Galois theory , Second edition, Research Notes in Mathematics, 1. A K Peters, Ltd., Wellesley,MA, 2008.[11] I.R. Shafarevich,
The embedding problem for split extensions , Dokl. Akad. Nauk SSSR 120 (1958), 1217-1219.[12] T. L. Smith,
Extra-special groups of order 32 as Galois groups,
Canad. J. Math. 46 (1994), no. 4, 886–896.
Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA
Email address : [email protected] URL ::