# Leopoldt's conjecture for totally real cubic and $S_3$-extensions

For an introduction to Leopoldt's conjecture, see part 3, Chapter X of "Cohomology of Number Fields", which is freely available here.

Write Leo($K$,$p$) for Leopoldt's conjecture for the number field $K$ at the prime $p$. If $K$ is an abelian extension of either $\mathbb{Q}$ or an imaginary quadratic field, then we know Leo($K$,$p$) for all primes $p$. (This is due to Ax and Brumer - see reference above for more details.)

Now let $K$ be a cubic extension of $\mathbb{Q}$. I am interested in Leo($K$,3). If $K/\mathbb{Q}$ is Galois then we are done. If $K$ has mixed signature, then its Galois closure $L$ is a totally complex $S_3$ extension of $\mathbb{Q}$ and thus is a cyclic extension of an imaginary quadratic field. Thus we know Leo($L$,3) and this implies Leo($K$,3) (Leopoldt's conjecture is inherited by subfields).

My question is: are there any cases for which Leo($K$,3) is known when $K$ is a totally real non-Galois cubic extension of $\mathbb{Q}$? By the above reasoning, it would suffice to find a totally real $S_3$ extension $F/\mathbb{Q}$ for which Leo($F$,3) is known.

• For an explicit field $K$ one could just check experimentally that the Leopoldt conjecture is true at 3, basically by computing the regulator and checking it's non-zero mod $3^m$ for some large $m$. – znt Nov 20 '16 at 16:27
• @znt Can you email me at the address on my website? – Henri Johnston Feb 8 '19 at 17:22

Out of curiosity, I thought I'd follow up on znt's suggestion.

Let $K = \mathbf{Q}(\alpha)$ where $\alpha^3 - 13\alpha + 7=0$. Then $K$ is totally real and non-Galois, and the prime 3 is totally inert in $K$. The unit group of $K$ is generated by $(u_1, u_2) = (\alpha^2 + 3\alpha - 2, \alpha^2 - 4\alpha + 2)$.

There are 3 embeddings of $K$ into the unramified cubic extension $L$ of $\mathbf{Q}_3$. Sage happens to like the polynomial $y^3 + 2y + 1$ as a defining polynomial for $L$; if $c$ is a root of this, then for $i = 0, 1, 2$ there's an embedding $\sigma_i$ for which $\sigma_i(\alpha) = c + i \bmod 3$.

Now one finds that the matrix with $i,j$ entry $\log \sigma_i(u_j)$ for $1 \le i, j \le 2$ has determinant

3^3 + 3^7 + 2*3^8 + 3^9 + 3^10 + 2*3^12 + 2*3^15 + 3^17 + 3^18 + 2*3^19 + 3^20 + O(3^21)


which is not zero. Hence Leopoldt's conjecture holds for this extension at 3.

• Thanks, David and znt! So in principle we can check Leo($K$,$p$) for any particular (totally real) number field $K$ and prime $p$ computationally. I'll hold off accepting your answer for the moment to see if there are any other answers that take a different approach. – Henri Johnston Nov 21 '16 at 11:58
• This article: Buchmann, Johannes; Sands, Jonathan, An algorithm for testing Leopoldt's conjecture, J. Number Theory 27 (1987), no. 1, 92–105 gives an algorithm that uses a reformulation of Leopoldt's conjecture that is more amenable to computation. However, as far as I know, there isn't an implementation of this algorithm in Magma or Sage. – Henri Johnston Nov 22 '16 at 12:32