Field trace

: coordinatewise complex conjugation converts the positive inner product into the trace pairing. This assertion does not require that conjugation preserve the number field itself.

Discriminant ideal ( $disc (S / R)$ )

 0  0

For the integral closure

S

of a Dedekind domain

R

in a finite separable extension, the discriminant ideal is locally generated by

det (Tr_{L / K} (α_{i} α_{j})),

(1)

where

(α_{i})

is a local

R

-basis of

S

. Intrinsically it is the image of

(det_{R} S)^{\otimes 2}

under the determinant of the trace pairing.

Relative discriminant ( $d_{L / K}$ )

 0  0

The relative discriminant

d_{L / K}

of a finite extension of number fields is the discriminant ideal of

O_{L}

over

O_{K}

. It satisfies

∣ d_{L} ∣ = ∣ d_{K} ∣^{[L : K]} N_{K / Q} (d_{L / K}),

(1)

and its prime divisors are exactly the ramified finite primes.

Norm of the different ideal ( $d_{E / K} = N_{E / K} (D_{E / K})$ )

 0  0

The relative discriminant of a finite extension of number fields is the relative norm of a fractional ideal applied to its different ideal. Consequently, at

p

v_{p} (d_{E / K}) = \sum_{P ∣ p} f_{P / p} d_{P},

(1)

where

f_{P / p}

is the residue-field degree and

d_{P}

is the different exponent. Over

Q

N (D_{E / Q}) = ∣ D_{E} ∣

Trace form of a number field ( $B (x, y) = Tr_{K / Q} (x y)$ )

 0  0

The trace form of a number field

K

is the symmetric bilinear form

B (x, y) = Tr_{K / Q} (x y) .

(1)

Nondegeneracy of the trace form of a number field

 0  0

For nonzero

x \in K

, choose

y = x^{- 1}

. Then

B (x, y) = Tr_{K / Q} (1) = [K : Q] \neq = 0,

(1)

so the trace form is nondegenerate.

Trace and norm integrality criterion for a quadratic extension

 0  0

In a quadratic extension

L / K

, an element is integral exactly when its relative trace and norm lie in

O_{K}

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Field trace can refer to different concepts depending on the context, so I'll outline a few possible interpretations: 1. **General Definition**: In a broad sense, a field trace could refer to a record or representation of observations or data collected from a specific field or area of study. This could be used in various disciplines, such as ecology, geography, or even data science.

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Field trace

Trace of an integral element over a normal domain

Trace pairing

Coefficient pairing for a monogenic algebra ( $λ (ab) = [X^{n - 1}] (ab mod g)$ )

Trace-dual lattice

Trace dual of a fractional ideal ( $b^{\lor} = D_{K / Q}^{- 1} b^{- 1}$ )

Discriminant ideal ( $disc (S / R)$ )

Relative discriminant ( $d_{L / K}$ )

Norm of the different ideal ( $d_{E / K} = N_{E / K} (D_{E / K})$ )

Trace form of a number field ( $B (x, y) = Tr_{K / Q} (x y)$ )

Nondegeneracy of the trace form of a number field

Trace and norm integrality criterion for a quadratic extension

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