Dodecic 3-adic fields (Q2890248)

This paper concerns the classification problem for degree \(n\) extensions of \(p\)-adic fields, focusing on the difficult case where \(p\) properly divides \(n\). In particular, the author provides a complete classification of degree 12 extensions of \(\mathbb{Q}_3\), the cases with composite \(n\leq 10\) and \(p|n\) having been dealt with earlier by \textit{J. W. Jones} and \textit{D. P. Roberts} [Lecture Notes in Computer Science 3076, 293--308 (2004; Zbl 1125.11356), J. Number Theory 128, No. 6, 1410-1429 (2008; Zbl 1140.11056)] and \textit{L. Gao} [J. Tianjin Norm. Univ., Nat. Sci. Ed. 26, No. 2, 35--37 (2006; Zbl 1142.11350)]. The situation for \(n=p\) was studied by \textit{S. Amano} [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18, 1-21 (1971; Zbl 0231.12019)].NEWLINENEWLINELet \(K|\mathbb{Q}_3\) be a degree 12 extension, and denote by \(G\) the Galois group of the Galois closure of \(K\). Then \(G\) is a transitive subgroup of the symmetric group \(S_{12}\), of which there are 265 up to isomorphism. The author begins by using the ramification filtration on the Galois group to show that only 45 of these are actually possibilities for \(G\). An examination of these 45 groups then shows that \(K|\mathbb{Q}_3\) must have a unique quartic subfield \(L|\mathbb{Q}_3\). But any quartic extension of \(\mathbb{Q}_3\) is at most tamely ramified, and standard results imply that there are exactly 5 such extensions up to isomorphism, for each of which the author provides a defining polynomial. Hence, in order to classify the degree 12 extensions \(K|\mathbb{Q}_3\), it just remains to classify the cubic extensions \(K|L\) for each of the 5 quartic extensions \(L|\mathbb{Q}_3\). For a given quartic field \(L\), there is a unique unramified cubic extension, while all other cubic extensions \(K|L\) are totally ramified, for which Amano's results cited above provide defining polynomials. Putting together the results for the 5 quartic fields, there are exactly 785 degree 12 extensions of \(\mathbb{Q}_3\).NEWLINENEWLINEIn the final section of the paper, the author determines the Galois group for each of these 785 extensions. His method is match the fields to the 45 possible groups determined previously by computing enough invariants, including the order of the centralizer of \(G\) in \(S_{12}\), the subfield Galois group content of \(G\), the parity of \(G\), and various resolvents.