The **discriminant** of a $p$-adic field $K$ is the square of the determinant of the matrix
\[
\left( \begin{array}{ccc}
\sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\
\vdots & & \vdots \\
\sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\
\end{array} \right)
\]
where $\sigma_1,..., \sigma_n$ are the embeddings of $K$ into an algebraic closure $\overline{\mathbb{Q}}_p$, and $\{\beta_1, \ldots, \beta_n\}$ is an integral basis for the ring of integers of $K$.

The discriminant of $K$ is an element of $\mathbb{Z}_p$ which is well-defined up to the square of a unit. Thus, it is of the form $p^c u$ where $u$ is a unit. The value $c$ is the discriminant exponent for $K$. Together with the discriminant root field of $K$, it determines the discriminant of $K$ (up to the square of a unit).

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-24 16:52:23

**Referred to by:**

**History:**(expand/hide all)

- 2020-10-24 16:52:23 by Andrew Sutherland (Reviewed)
- 2020-10-24 16:33:59 by Andrew Sutherland
- 2018-05-23 14:43:50 by John Cremona (Reviewed)

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