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The norm and trace of a finite field extension
Definition
Let be a finite field extension, so is a finite-dimensional -vector space of degree (The degree of a finite field extension). For , let
the -linear operator of multiplication by .
The norm and trace of from to are
where determinant and trace are those of the published linear-operator notions (The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and on the zero space, The basis-independent trace of an endomorphism of a finite-dimensional vector space).
Because is a field, is the zero operator exactly when and is an automorphism exactly when . Later items identify these two quantities with the classical embedding formulas and the trace form.
Depends on
- The degree $[K:F]=\dim_F K$ of a finite field extension
- The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
Used by
- Every nontrivial number field has a ramified finite prime Corollary
- The abstract residue computes the coefficient-trace residue at every closed point Corollary
- Trace and norm of an algebraic integer Corollary
- Discriminant of a basis and order Definition
- Residue of a rational differential at a separable closed point Definition
- The trace form (x,y)↦ Tr_K/F(xy) of a finite extension Definition
- Class group of Q(sqrt -5) Example
- Class group of Q(sqrt 10) Example
- A nonzero global dual section detects a cohomology class Lemma
- A number-field unit is exactly an algebraic integer of norm plus or minus one Lemma
- Annihilators of regular sections under the local residue pairing Lemma
- Equivalent formulations of the Hilbert symbol Lemma
- Residues of exact differentials vanish Lemma
- The residue is independent of the uniformizer Lemma
- Normalization of the trace for Serre duality on a curve Remark
- Field norm and trace agree with the determinant and trace of multiplication by an element Theorem
- Hasse-Minkowski for ternary forms over Q Theorem
- Hilbert's theorem 90 for a finite cyclic extension Theorem
- Kronecker root-of-unity criterion Theorem
- Norm and trace from embeddings, with the inseparable exponent in the norm formula Theorem
- Norm is multiplicative, trace is F-linear, and both are transitive in towers Theorem
- Ramification is detected by the number-field discriminant Theorem
- The ring of integers has rank the degree Theorem
Dependency tree · two levels
9 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
- J. S. Milne, Fields and Galois Theory, v5.10, Chapter 6, Section 5 (standard reference, not scraped)
- B. Conrad, Norm and trace, Sections 1-3 (standard reference, not scraped)