The North Carolina Journal of Mathematics and Statistics

Constructing Galois 2-extensions of the 2-adic Numbers

Chad Awtrey, James R. Beuerle, Jade Schrader

Abstract


Let Q_2 denote the field of 2-adic numbers, and let G be a group of order 2^n for some positive integer n. We provide an implementation in the software program GAP of an algorithm due to Yamagishi that counts the number of nonisomorphic Galois extensions K/Q_2 whose Galois group is G. Furthermore, we describe an algorithm for constructing defining polynomials for each such extension by considering quadratic extensions of Galois 2-adic fields of degree 2^{n−1}. While this method does require that some extensions be discarded, we show that this approach considers far fewer extensions than the best general construction algorithm currently known, which is due to Pauli- Sinclair based on the work of Monge. We end with an application of our approach to completely classify all Galois 2-adic fields of degree 16, including defining polynomials, ramification index, residue degree, valuation of the discriminant, and Galois group.

Keywords


2-groups, extension fields, Galois group, 2-adic

Full Text:

PDF

References


Chad Awtrey, Brett Barkley, Nicole E. Miles, Christopher Shill, and Erin Strosnider, Degree 12 2-adic fields with automorphism group of order 4, Rocky Mountain J. Math. 45 (2015), no. 6, 1755–1764.

Chad Awtrey, John Johnson, Jonathan Milstead, and Brian Sinclair, Groups of order 16 as Galois groups over the 2-adic numbers, Int. J. Pure Appl. Math. 103 (2015), no. 4, 781–795.

Chad Awtrey, Nicole Miles, Jonathan Milstead, Christopher Shill, and Erin Strosnider, Degree 14 2-adic fields, Involve 8 (2015), no. 2, 329–336.

Chad Awtrey, Nicole Miles, Christopher Shill, and Erin Strosnider, Computing Galois groups of degree 12 2-adic fields with trivial automorphism group, JP J. Algebra Number Theory Appl. 38 (2016), no. 5, 457–471.

Chad Awtrey and Christopher R. Shill, Galois groups of degree 12 2-adic fields with automorphism group of order 6 and 12, Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics, vol. 64, Springer, New York, 2013, pp. 55–65.

Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993).

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6.5, 2013.

John W. Jones and David P. Roberts, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80–97.

John W. Jones and David P. Roberts, Octic 2-adic fields, J. Number Theory 128 (2008), no. 6, 1410–1429.

John P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132.

Maurizio Monge, Determination of the number of isomorphism classes of extensions of a p-adic field, J. Number Theory 131 (2011), no. 8, 1429–1434.

Maurizio Monge, A family of Eisenstein polynomials generating totally ramified extensions, identification of extensions and construction of class fields, Int. J. Number Theory 10 (2014), no. 7, 1699–1727.

Sebastian Pauli and Xavier-Francois Roblot, On the computation of all extensions of a p-adic field of a given degree, Math. Comp. 70 (2001), no. 236, 1641–1659.

Sebastian Pauli and Brian Sinclair, Enumerating extensions of pi-adic fields with given invariants, Int. J. Number Theory 13 (2017), no. 8, 2007–2038.

Masakazu Yamagishi, On the number of Galois p-extensions of a local field, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2373–2380.