The North Carolina Journal of Mathematics and Statistics

Constructing Galois 2-extensions of the 2-adic Numbers

Chad Awtrey, James R. Beuerle, Jade Schrader


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.


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

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