For an introduction to Leopoldt's conjecture, see part 3, Chapter X of "Cohomology of Number Fields", which is freely available [here](https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/).


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.



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


