ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-26

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For $Q (\sqrt{d}) / Q$ , the embedding formulas match the determinant and trace of multiplication

Example

Let $d \in Q$ be nonsquare, let $K = Q (\sqrt{d})$ , and write $α = a + b \sqrt{d}$ with $a, b \in Q$ . Then

$N_{K / Q} (α) = a^{2} - d b^{2}, {Tr}_{K / Q} (α) = 2 a .$

In the basis $(1, \sqrt{d})$ , multiplication by $α$ has matrix

$(\begin{array}{cc} a & b d \\ b & a \end{array}),$

so the field norm and trace agree with the determinant and trace of that matrix.

Facts & Assumptions

Given: The quadratic extension $K = Q (\sqrt{d})$ with $d$ nonsquare, and the element $α = a + b \sqrt{d}$ .

[L1]

Norm and trace are the product and sum of the embeddings in the separable case (Norm and trace from embeddings, with the inseparable exponent in the norm formula).

[L2]

Field norm and trace agree with determinant and trace of multiplication by the element (Field norm and trace agree with the determinant and trace of multiplication by an element).

Verification

technique · direct

1.1L1algebra

The two $Q$ -embeddings of $K$ send $\sqrt{d}$ to $\pm \sqrt{d}$ , so [L1] gives $N_{K / Q} (α) = (a + b \sqrt{d}) (a - b \sqrt{d}) = a^{2} - d b^{2},$ and ${Tr}_{K / Q} (α) = (a + b \sqrt{d}) + (a - b \sqrt{d}) = 2 a .$

2.1L2step 1.1algebra∎

Multiplication by $α$ sends $1 \mapsto a + b \sqrt{d}, \sqrt{d} \mapsto b d + a \sqrt{d},$ so its matrix in the basis $(1, \sqrt{d})$ is the displayed matrix. That matrix has determinant $a^{2} - d b^{2}$ and trace $2 a$ , agreeing with step 1.1 as [L2] predicts.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

B. Conrad, Norm and trace, Section 1 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Section 5 (standard reference, not scraped)

For Q(d)/QQ(d​)/Q, the embedding formulas match the determinant and trace of multiplication