DEGREE 14 EXTENSIONS OF Q7
JIM BROWN, ROBERT CASS, RODNEY KEATON, SALVATORE PARENTI,AND DANIEL SHANKMANAbstract. We compute all degree 14 extensions of Q7 up to isomorphism, and find that there are 654 such extensions. Additionally, we
compute several invariants of these extensions in order to classify theassociated Galois group of the Galois closure of each extension.

1. Introduction
For a prime p and a positive integer n, it is well-known that there are onlyﬁnitely many degree n ﬁeld extensions of the ﬁeld Qp of p-adic numbers.
When p - n or p = n, all of the extensions of Qp have been classiﬁed anddata associated to these extensions is stored in an online database of local
ﬁelds created by Jones and Roberts ([7]). When p properly divides n, theproblem of classifying these extensions becomes much more complicated. In
this case, such extensions have been classiﬁed completely for all n ≤ 12.
In this paper, we focus on the case n = 14 and p = 7. We use methods
established by S. Pauli and implemented in the computer algebra system
MAGMA ([4]) to compute deﬁning polynomials for each of these extensions
up to isomorphism. Using ramiﬁcation groups, we obtain a list which includes all possible Galois groups of an irreducible degree 14 polynomial overQ7 . Employing computational methods motivated by those used in [3] toclassify degree 12 extensions of Q3 , we compute several invariants of theseextensions to determine the Galois groups of their deﬁning polynomials.
Throughout this paper we ﬁx an algebraic closure Q7 and work in thisalgebraic closure.
2. Defining Polynomials
In his thesis, Pauli developed methods to compute deﬁning polynomials
for extensions of local ﬁelds ([8]). Using the implementation of his techniques
in MAGMA, we compute a list of deﬁning polynomials for all extensionsof Q7 . We obtain 1158 such irreducible polynomials. Sorting the extensions deﬁned by these polynomials into isomorphism classes can be doneusing Panayi’s p-adic root ﬁnding algorithm, which is also implemented inMAGMA. Two degree n extensions of Qp are isomorphic if and only if their2010 Mathematics Subject Classiﬁcation. Primary 11S15, 11S20.
Key words and phrases. Local fields, Galois groups.
The authors were supported by NSF DMS-1156734.
12J. BROWN, R. CASS, R. KEATON, S. PARENTI, AND D. SHANKMAN

deﬁning polynomials share a common root. Table 1 gives the counts for theextensions of Q7 . Here e is the ramiﬁcation degree, c is the discriminant
exponent, #Ke,c is the total number of extensions as counted by Krasner’s
mass formula, and #Qe,c7 is the number of non-isomorphic extensions.
Table 1. Counts for Extensions of Q7e127c0714161820
22
2426
14 8
91011
12
13
15
16
17
18
19
20
21

# Ke,c
1
2
336
336
336
336
336
336
343
84
84
84
84
84
84
588
588
588
588
588
588
686

# Qe,c
7
1
2
27
27
27
27
27
54
28
6
12
6
12
6
18
48
42
48
42
96
4256=654

Some of the polynomials obtained have coeﬃcients with many digits. Inorder to obtain polynomials with coeﬃcients more amenable to computation,
we randomly generated degree 14 polynomials with small integer coeﬃcients.
Each random polynomial was checked for irreducibility using Panayi’s root
ﬁnding algorithm. If the random polynomial was irreducible, we found theunique polynomial f in our list of 654 polynomials deﬁning the same isomorphism class of extensions. If the random polynomial had coeﬃcients withfewer digits, we let it replace f in our list of 654 polynomials. One can see[5] for the complete list of all 654 polynomials.
3. Ramification Groups
In this section we introduce ramiﬁcation groups and use them to gain
information about the Galois groups of the Galois closures of degree 14DEGREE 14 EXTENSIONS OF Q73extensions of Q7 . For more facts about ramiﬁcation groups one can see [2]or [9].
Let L/Qp be a Galois extension and set G = Gal(L/Qp ). Let νL be thediscrete valuation on L and let OL be the valuation ring. For each integeri ≥ 1 we deﬁne the ith ramiﬁcation group of G to be
Gi = {σ ∈ G : νL (σ(x) − x) ≥ i + 1

for every x ∈ OL }.

Note that G−1 = G and that the ramiﬁcation groups form a decreasingﬁltration on G which is eventually trivial. Additionally, G/G0 is isomorphic
to the Galois group of the residue ﬁeld extension. The following lemma gives
some structural information about the ramiﬁcation groups.
Lemma 1 (Corl. 4.1.3, [2]). Let L/Qp be a Galois extension with ϖ auniformizer for L and let G = Gal(L/Qp ). Let Ui = ⟨1 + (ϖi )⟩ and let U0
be the group of units of OL . Then
(1) For i ≥ 0, Gi /Gi+1 is isomorphic to a subgroup of Ui /Ui+1 and hence
is abelian.
(2) G0 /G1 is cyclic with order coprime to p.
(3) For i ≥ 0, Gi /Gi+1 is a direct product of cyclic groups of order p.
(4) G0 is the semi-direct product of a cyclic group of order coprime to p
and a normal subgroup which is a p-group.
(5) G and G0 are solvable.
In our case we have a degree 14 extension K/Q7 with G = Gal(K gal /Q7 )and with K gal the Galois closure of K in our chosen algebraic closure of
Q7 . Thus G is one of the 63 solvable transitive subgroups of S14 . Also, G
must contain a solvable normal subgroup G0 such that G/G0 is cyclic andhas order dividing 14 since G/G0 is isomorphic to the Galois group of theresidue ﬁeld extension. Moreover, G0 /G1 is cyclic and has order dividing
7[G:G0 ] − 1. This last statement follows from part (1) of the lemma and thefact that U0 /U1 is isomorphic to the multiplicative group of the residue ﬁeldof L. Using GAP ([6]), computations on the 63 transitive subgroups of S14
show that there only 17 possible Galois groups that can be associated to adegree 14 extension of Q7 . Listed using the T notation in the LMFDB ([1]),these 17 groups are given by 14T n where n is one of
1, 2, 3, 4, 5, 7, 8, 12, 13, 14, 15, 20, 22, 23, 24, 25, 32.
However, we can eliminate three of these groups. The groups 14T 15, 14T 22,
and 14T 25 each have the low degree resolvent 3T 2. This means that any
extension of Q7 whose Galois group is one of these three groups must containa degree 3 subﬁeld whose associated Galois group is 3T 2. This is impossible
since none of the degree 3 extensions of Q7 have the associated Galois group
3T 2. Thus we can eliminate these three groups, leaving us with the following
list of 14 possible Galois groups 14T n where n is one of
1, 2, 3, 4, 5, 7, 8, 12, 13, 14, 20, 23, 24, 32.

4

J. BROWN, R. CASS, R. KEATON, S. PARENTI, AND D. SHANKMAN

4. Galois groups
We now compute the Galois group of each of our 654 deﬁning polynomials. To accomplish this task, we compute enough invariants to uniquelydetermine the Galois group of each polynomial. Following the approach usedin [3], we ﬁrst summarize three of the more basic invariants that we use.
The ﬁrst of these invariants is the subﬁeld content of an extension K/Q7 ,which is the list of the Galois groups of the Galois closures of the proper
nontrivial subﬁelds of K. On the group theory side, the subﬁeld content
for a degree 14 ﬁeld extension associated with any of our 14 possible Galois
groups has been computed and is listed in the LMFDB. We list the subﬁeld
content associated to these Galois groups in Table 2, and immediately noticethat all degree 14 extensions of Q7 must have a unique quadratic subﬁeld.
In order to compute the subﬁeld content of a degree 14 extension K/Q7 ,we must only check if any of the ﬁnitely many degree 2 extensions of Q7or degree 7 extensions of Q7 , which have been completely classiﬁed, aresubﬁelds of K. Such a degree 2 or degree 7 extension of Q7 is a subﬁeld of
K if and only if the deﬁning polynomial of the degree 2 or 7 extension has aroot in K. We already know that K must have a unique quadratic subﬁeld,
but we will make use of the deﬁning polynomial of that subﬁeld later so westill determine the quadratic subﬁeld here.
We also make use of the order of the centralizer of the Galois group inS14 , which is equal to the order of the automorphism group Aut(K/Q7 ).
This automorphism group can be computed easily as MAGMA has built-infunctionality for computing Aut(K/Q7 ). The centralizer order of our 14
possible Galois groups is listed in Table 2 under the heading C.O.
Additionally, we use the parity of the the Galois groups, which is +1 ifG ⊆ A14 and -1 otherwise. The parity of a deﬁning polynomial f is +1 if its
discriminant is a square in Q7 and -1 otherwise. To compute the parity ofthe deﬁning polynomials, we use the algorithm outlined in [2]. We list theparity of each possible Galois group in Table 2.
As these invariants are not enough to distinguish all the groups, it isnecessary to use two resolvent polynomials to uniquely determine each Galois
group. The ﬁrst of these is the following degree 91 linear absolute resolvent∏f91 (x) =(x − (αi + αj ))1≤i<j≤14

where the αi are the roots of the deﬁning polynomial f . This can be computed in terms of resultants via the formula()
Resultanty (f (y), f (x − y)) 1/2
f91 (x) =.
214 f (x/2)
If the resolvent polynomial f91 (x) is square-free, then degrees of the irreducible factors of f91 (x) factored over Q7 correspond to the orbit lengths ofthe component-wise action of G on ordered pairs (a1 , a2 ) where a1 ̸= a2 anda1 , a2 ∈ {1, . . . , 14}. These orbit lengths are listed for some of the Galois

DEGREE 14 EXTENSIONS OF Q7

5groups in Table 2 under the heading O.L. If f91 (x) is not square-free, weapply a Tschirnhausen transformation to the deﬁning polynomial f to geta new irreducible polynomial deﬁning the same extension as f . Eventually
we get a resolvent that is square-free.
We also use a quartic relative resolvent f4 (x), analogous to one used in[3], which relies on the unique quadratic subﬁeld. Let f deﬁne a degree 14
extension K/Q7 and let F be the unique quadratic subﬁeld of K. Let g be
any of the 7 quadratic polynomials obtained by factoring f over K. Thenf4 (x) is the norm of x2 − disc(g(x)) down to Q7 . That is,
f4 (x) = (x2 − disc(g(x)))(x2 − σ(disc(g(x))))where σ ∈ Gal(F/Q7 ) is the unique nontrivial automorphism. To aid incomputation, note that the coeﬃcients of f4 (x) are symmetric polynomials
in the roots of the deﬁning polynomial of K. We are interested in the Galois
group of f4 (x), which can easily be computed using MAGMA as the degree
is suﬃciently small. Table 2 lists the Galois groups of f4 (x) for extensions
having some of our 14 possible Galois groups. These Galois groups werecomputed using polynomials with integer coeﬃcients deﬁning number ﬁelds
and having the given Galois group. In one case f4 (x) is not irreducible,
which is why the Galois group has only two elements.
Table 2. Possible Galois Groups for p = 7Label (14T)123
457812131420
232432Subﬁelds
7T1, 2T1
7T2, 2T1
7T2, 2T1
7T4, 2T1
7T3, 2T1
7T4, 2T1
2T1
2T1
2T1
2T1
2T1
2T1
2T12T1C.O.
14
142222
71
111
1
1
1Parity-1-1-1-1-1-1-1
1-1-1-1
1-1-1O.L.

Gal(f4 )[7, 212 , 42][7,422 ][143 , 49][143 , 49][42,49][143 , 49][42,49][42,49][42,49]

4T2
2T1
4T3
4T1
4T2
4T3

From Table 2 it is clear that the Galois groups 14T 1, 14T 2, 14T 3, 14T 5,and 14T 8 are uniquely determined by their subﬁeld content, centralizer order, and parity. By also making use of the degree 91 absolute resolvent, weare able to distinguish the Galois groups 14T 4, 14T 7, and 14T 12. Finally,by using the resolvent f4 (x) we are able to identify the remaining Galois
groups 14T 13, 14T 14, 14T 20, 14T 23, 14T 24, and 14T 32.

6

J. BROWN, R. CASS, R. KEATON, S. PARENTI, AND D. SHANKMAN

We conclude by giving a count of the number of degree 14 extensions ofQ7 having each of the 14 Galois groups. Table 3 gives this data.

Table 3. Counts for degree 14 extensions of Q7 by Galois groupGalois Group (14T)
123457812131420232432

Number of Extensions
243631246272
89931664114128

References[1] LMFDB. http://www.lmfdb.org/. Accessed May 29, 2013.
[2] C. Awtrey. Dodecic local ﬁelds. PhD thesis, Arizona State University, Tempe, AZ,
2010.
[3] C. Awtrey. Dodecic 3-adic Fields. Int. J. Num. Th., 8:933–944, 2012.
[4] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system.
I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computationalalgebra and number theory (London, 1993).
[5] J.
Brown,R.
Cass,R.
Keaton,S.
Parenti,andD.
Shankman.
Datatables
fordegree
14
extensionsofQ7 .
http://www.ces.clemson.edu/~jimlb/ResearchPapers/REU2013data.pdf. Accessed
August 31, 2013.
[6] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.5.5, 2012.
[7] J. Jones and D. Roberts. A database of local fields. J. Symbolic Comput., 41:80–97,2006.
[8] Sebastian Pauli. Eﬃcient Enumeration of Extensions of Local Fields with Bounded
Discriminant. PhD thesis, Concordia University, 2001.
[9] J-P. Serre. Local ﬁelds, volume 67 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1979.

DEGREE 14 EXTENSIONS OF Q7

7

Department of Mathematical Sciences, Clemson University, Clemson, SC
29634
E-mail address: jimlb@clemson.edu
Department of Mathematics, University of Kentucky, Lexington, KY 40506
E-mail address: robert.cass@uky.edu
Department of Mathematical Sciences, Clemson University, Clemson, SC
29634
E-mail address: rkeaton@clemson.edu
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
E-mail address: sparenti@umich.edu
Department of Mathematics, University of Tennessee, Knoxville, TN 37996
E-mail address: dshankma@utk.edu