TheoremStatement: Literature-sourcedProof: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Embedding determinant formula

Statement

Let $σ_{1}, \dots, σ_{n}$ be the $n$ distinct embeddings $K \to C$ . For every ordered $Q$ -basis $α_{1}, \dots, α_{n}$ of $K$ , $disc (α_{1}, \dots, α_{n}) = \det (σ_{i} (α_{j}))^{2},$ and the determinant is nonzero.

Facts & Assumptions

Given: An ordered basis of the number field.

[F1]

In a separable extension, trace is the sum over the embeddings (Norm and trace from embeddings, with the inseparable exponent in the norm formula).

[F2]

Distinct field embeddings are linearly independent as characters (Dedekind's linear independence theorem for distinct characters).

Proof

technique · direct

1.1F1given

With $A = (σ_{i} (α_{j}))$ , [F1] gives $G = A^{t} A$ for the trace Gram matrix.

1.2F2given

If $A$ were singular, a nonzero linear combination of its rows would vanish on the basis and hence on every element of $K$ , contradicting [F2]. Thus $\det A \neq 0$ .

2.1step 1.1step 1.2algebra∎

Taking determinants in step 1.1 gives $\det G = \det (A)^{2}$ , and step 1.2 gives the asserted nonvanishing.

Depends on

Used by

Unscaled Minkowski embedding Definition
Integral basis and discriminant of a coprime-discriminant compositum Lemma
Covolume of an integral ideal lattice Theorem
Number-field discriminant is well-defined and nonzero Theorem
Number-field integer rings and ideals are full lattices Theorem
Power-basis and polynomial discriminants Theorem
Signed discriminant of a cyclotomic field Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Milne, Proposition 2.26 (standard reference, not scraped)