# Field trace

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Mathematical function

For other uses, see [Trace (disambiguation)](/source/Trace_(disambiguation)).

In [mathematics](/source/Mathematics), the **field trace** is a particular [function](/source/Function_(mathematics)) defined with respect to a [finite](/source/Finite_extension) [field extension](/source/Field_extension) *L*/*K*, which is a [*K*-linear map](/source/Linear_map) from *L* onto *K*.

## Definition

Let *K* be a [field](/source/Field_(mathematics)) and *L* a finite extension (and hence an [algebraic extension](/source/Algebraic_extension)) of *K*. *L* can be viewed as a [vector space](/source/Vector_space) over *K*. Multiplication by *α*, an element of *L*,

- m α : L → L given by m α ( x ) = α x {\displaystyle m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x} ,

is a *K*-[linear transformation](/source/Linear_transformation) of this vector space into itself. The *trace*, **Tr***L*/*K*(*α*), is defined as the [trace](/source/Trace_(linear_algebra)) (in the [linear algebra](/source/Linear_algebra) sense) of this linear transformation.[1]

For *α* in *L*, let *σ*1(*α*), ..., *σ**n*(*α*) be the [roots](/source/Root_of_a_polynomial) (counted with multiplicity) of the [minimal polynomial](/source/Minimal_polynomial_(field_theory)) of *α* over *K* (in some extension field of *K*). Then

- Tr L / K ⁡ ( α ) = [ L : K ( α ) ] ∑ j = 1 n σ j ( α ) . {\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha ).}

If *L*/*K* is [separable](/source/Separable_extension) then each root appears only once[2] (however this does not mean the coefficient above is one; for example if *α* is the identity element 1 of *K* then the trace is [*L*:*K*] times 1).

More particularly, if *L*/*K* is a [Galois extension](/source/Galois_extension) and *α* is in *L*, then the trace of *α* is the sum of all the [Galois conjugates](/source/Galois_conjugate) of *α*,[1] i.e.,

- Tr L / K ⁡ ( α ) = ∑ σ ∈ Gal ⁡ ( L / K ) σ ( α ) , {\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\sum _{\sigma \in \operatorname {Gal} (L/K)}\sigma (\alpha ),}

where Gal(*L*/*K*) denotes the [Galois group](/source/Galois_group) of *L*/*K*.

## Example

Let L = Q ( d ) {\displaystyle L=\mathbb {Q} ({\sqrt {d}})} be a [quadratic extension](/source/Quadratic_extension) of Q {\displaystyle \mathbb {Q} } . Then a [basis](/source/Basis_(linear_algebra)) of L / Q {\displaystyle L/\mathbb {Q} } is { 1 , d } . {\displaystyle \{1,{\sqrt {d}}\}.} If α = a + b d {\displaystyle \alpha =a+b{\sqrt {d}}} then the [matrix](/source/Matrix_(mathematics)) of m α {\displaystyle m_{\alpha }} is:

- [ a b d b a ] {\displaystyle \left[{\begin{matrix}a&bd\\b&a\end{matrix}}\right]} ,

and so, Tr L / Q ⁡ ( α ) = [ L : Q ( α ) ] ( σ 1 ( α ) + σ 2 ( α ) ) = 1 × ( σ 1 ( α ) + σ 1 ¯ ( α ) ) = a + b d + a − b d = 2 a {\displaystyle \operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a} .[1] The minimal polynomial of *α* is *X*2 − 2*a* *X* + (*a*2 − *db*2).

## Properties of the trace

Several properties of the trace function hold for any finite extension.[3]

The trace Tr*L*/*K* : *L* → *K* is a *K*-[linear map](/source/Linear_map) (a *K*-linear functional), that is

- Tr L / K ⁡ ( α a + β b ) = α Tr L / K ⁡ ( a ) + β Tr L / K ⁡ ( b ) for all α , β ∈ K {\displaystyle \operatorname {Tr} _{L/K}(\alpha a+\beta b)=\alpha \operatorname {Tr} _{L/K}(a)+\beta \operatorname {Tr} _{L/K}(b){\text{ for all }}\alpha ,\beta \in K} .

If *α* ∈ *K* then Tr L / K ⁡ ( α ) = [ L : K ] α . {\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .}

Additionally, trace behaves well in [towers of fields](/source/Tower_of_fields): if *M* is a finite extension of *L*, then the trace from *M* to *K* is just the [composition](/source/Function_composition) of the trace from *M* to *L* with the trace from *L* to *K*, i.e.

- Tr M / K = Tr L / K ∘ Tr M / L {\displaystyle \operatorname {Tr} _{M/K}=\operatorname {Tr} _{L/K}\circ \operatorname {Tr} _{M/L}} .

## Finite fields

Let *L* = GF(*q**n*) be a finite extension of a [finite field](/source/Finite_field) *K* = GF(*q*). Since *L*/*K* is a [Galois extension](/source/Galois_extension), if *α* is in *L*, then the trace of *α* is the sum of all the [Galois conjugates](/source/Galois_conjugate) of *α*, i.e.[4]

- Tr L / K ⁡ ( α ) = α + α q + ⋯ + α q n − 1 . {\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\alpha +\alpha ^{q}+\cdots +\alpha ^{q^{n-1}}.}

In this setting we have the additional properties:[5]

- Tr L / K ⁡ ( a q ) = Tr L / K ⁡ ( a ) for a ∈ L {\displaystyle \operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L} .

- For any α ∈ K {\displaystyle \alpha \in K} , there are exactly q n − 1 {\displaystyle q^{n-1}} elements b ∈ L {\displaystyle b\in L} with Tr L / K ⁡ ( b ) = α {\displaystyle \operatorname {Tr} _{L/K}(b)=\alpha } .

*Theorem*.[6] For *b* ∈ *L*, let *F**b* be the map a ↦ Tr L / K ⁡ ( b a ) . {\displaystyle a\mapsto \operatorname {Tr} _{L/K}(ba).} Then *F**b* ≠ *F**c* if *b* ≠ *c*. Moreover, the *K*-linear transformations from *L* to *K* are exactly the maps of the form *F**b* as *b* varies over the field *L*.

When *K* is the [prime subfield](/source/Prime_subfield) of *L*, the trace is called the *absolute trace* and otherwise it is a *relative trace*.[4]

### Application

A [quadratic equation](/source/Quadratic_equation), *ax*2 + *bx* + *c* = 0 with *a* ≠ 0, and coefficients in the finite field GF ⁡ ( q ) = F q {\displaystyle \operatorname {GF} (q)=\mathbb {F} _{q}} has either 0, 1 or 2 roots in GF(*q*) (and two roots, counted with multiplicity, in the quadratic extension GF(*q*2)). If the [characteristic](/source/Characteristic_(algebra)) of GF(*q*) is [odd](/source/Parity_(mathematics)), the [discriminant](/source/Discriminant) Δ = *b*2 − 4*ac* indicates the number of roots in GF(*q*) and the classical [quadratic formula](/source/Quadratic_formula) gives the roots. However, when GF(*q*) has [even](/source/Parity_(mathematics)) characteristic (i.e., *q* = 2*h* for some positive [integer](/source/Integer) *h*), these formulas are no longer applicable.

Consider the quadratic equation *ax*2 + *bx* + c = 0 with coefficients in the finite field GF(2*h*).[7] If *b* = 0 then this equation has the unique solution x = c a {\displaystyle x={\sqrt {\frac {c}{a}}}} in GF(*q*). If *b* ≠ 0 then the substitution *y* = *ax*/*b* converts the quadratic equation to the form:

- y 2 + y + δ = 0 , where δ = a c b 2 . {\displaystyle y^{2}+y+\delta =0,{\text{ where }}\delta ={\frac {ac}{b^{2}}}.}

This equation has two solutions in GF(*q*) [if and only if](/source/If_and_only_if) the absolute trace Tr G F ( q ) / G F ( 2 ) ⁡ ( δ ) = 0. {\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(\delta )=0.} In this case, if *y* = *s* is one of the solutions, then *y* = *s* + 1 is the other. Let *k* be any element of GF(*q*) with Tr G F ( q ) / G F ( 2 ) ⁡ ( k ) = 1. {\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(k)=1.} Then a solution to the equation is given by:

- y = s = k δ 2 + ( k + k 2 ) δ 4 + … + ( k + k 2 + … + k 2 h − 2 ) δ 2 h − 1 . {\displaystyle y=s=k\delta ^{2}+(k+k^{2})\delta ^{4}+\ldots +(k+k^{2}+\ldots +k^{2^{h-2}})\delta ^{2^{h-1}}.}

When *h* = 2*m'* + 1, a solution is given by the simpler expression:

- y = s = δ + δ 2 2 + δ 2 4 + … + δ 2 2 m . {\displaystyle y=s=\delta +\delta ^{2^{2}}+\delta ^{2^{4}}+\ldots +\delta ^{2^{2m}}.}

## Trace form

When *L*/*K* is separable, the trace provides a [duality theory](/source/Duality_theory) via the **trace form**: the map from *L* × *L* to *K* sending (*x*, *y*) to Tr*L*/*K*(*xy*) is a [nondegenerate](/source/Nondegenerate_form), [symmetric bilinear form](/source/Symmetric_bilinear_form) called the trace form. If *L*/*K* is a Galois extension, the trace form is invariant with respect to the Galois group.

The trace form is used in [algebraic number theory](/source/Algebraic_number_theory) in the theory of the [different ideal](/source/Different_ideal).

The trace form for a finite degree field extension *L*/*K* has non-negative [signature](/source/Signature_(quadratic_form)) for any [field ordering](/source/Field_ordering) of *K*.[8] The [converse](/source/Converse_(logic)), that every [Witt equivalence](/source/Witt_ring_(forms)) class with non-negative signature contains a trace form, is true for [algebraic number fields](/source/Algebraic_number_field) *K*.[8]

If *L*/*K* is an [inseparable extension](/source/Inseparable_extension), then the trace form is identically 0.[9]

## See also

- [Field norm](/source/Field_norm)

- [Reduced trace](/source/Reduced_trace)

## Notes

1. ^ [***a***](#cite_ref-ROT940_1-0) [***b***](#cite_ref-ROT940_1-1) [***c***](#cite_ref-ROT940_1-2) [Rotman 2002](#CITEREFRotman2002), p. 940

1. **[^](#cite_ref-2)** [Rotman 2002](#CITEREFRotman2002), p. 941

1. **[^](#cite_ref-3)** [Roman 2006](#CITEREFRoman2006), p. 151

1. ^ [***a***](#cite_ref-LN54_4-0) [***b***](#cite_ref-LN54_4-1) [Lidl & Niederreiter 1997](#CITEREFLidlNiederreiter1997), p.54

1. **[^](#cite_ref-5)** [Mullen & Panario 2013](#CITEREFMullenPanario2013), p. 21

1. **[^](#cite_ref-LN56_6-0)** [Lidl & Niederreiter 1997](#CITEREFLidlNiederreiter1997), p.56

1. **[^](#cite_ref-7)** [Hirschfeld 1979](#CITEREFHirschfeld1979), pp. 3-4

1. ^ [***a***](#cite_ref-L38_8-0) [***b***](#cite_ref-L38_8-1) Lorenz (2008) p.38

1. **[^](#cite_ref-9)** [Isaacs 1994](#CITEREFIsaacs1994), p. 369 as footnoted in [Rotman 2002](#CITEREFRotman2002), p. 943

## References

- Hirschfeld, J.W.P. (1979), [*Projective Geometries over Finite Fields*](https://archive.org/details/projectivegeomet0000hirs), Oxford Mathematical Monographs, Oxford University Press, [ISBN](/source/ISBN_(identifier)) [0-19-853526-0](https://en.wikipedia.org/wiki/Special:BookSources/0-19-853526-0)

- Isaacs, I.M. (1994), *Algebra, A Graduate Course*, Brooks/Cole Publishing

- Lidl, Rudolf; [Niederreiter, Harald](/source/Harald_Niederreiter) (1997) [1983], [*Finite Fields*](https://archive.org/details/finitefields0000lidl_a8r3), Encyclopedia of Mathematics and its Applications, vol. 20 (Second ed.), [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [0-521-39231-4](https://en.wikipedia.org/wiki/Special:BookSources/0-521-39231-4), [Zbl](/source/Zbl_(identifier)) [0866.11069](https://zbmath.org/?format=complete&q=an:0866.11069)

- Lorenz, Falko (2008). *Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics*. Springer. [ISBN](/source/ISBN_(identifier)) [978-0-387-72487-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-72487-4). [Zbl](/source/Zbl_(identifier)) [1130.12001](https://zbmath.org/?format=complete&q=an:1130.12001).

- Mullen, Gary L.; Panario, Daniel (2013), *Handbook of Finite Fields*, CRC Press, [ISBN](/source/ISBN_(identifier)) [978-1-4398-7378-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4398-7378-6)

- Roman, Steven (2006), *Field theory*, Graduate Texts in Mathematics, vol. 158 (Second ed.), Springer, Chapter 8, [ISBN](/source/ISBN_(identifier)) [978-0-387-27677-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-27677-9), [Zbl](/source/Zbl_(identifier)) [1172.12001](https://zbmath.org/?format=complete&q=an:1172.12001)

- Rotman, Joseph J. (2002), *Advanced Modern Algebra*, Prentice Hall, [ISBN](/source/ISBN_(identifier)) [978-0-13-087868-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-087868-7)

## Further reading

- Conner, P.E.; Perlis, R. (1984). *A Survey of Trace Forms of Algebraic Number Fields*. Series in Pure Mathematics. Vol. 2. World Scientific. [ISBN](/source/ISBN_(identifier)) [9971-966-05-0](https://en.wikipedia.org/wiki/Special:BookSources/9971-966-05-0). [Zbl](/source/Zbl_(identifier)) [0551.10017](https://zbmath.org/?format=complete&q=an:0551.10017).

- Section VI.5 of [Lang, Serge](/source/Serge_Lang) (2002), *[Algebra](/source/Algebra_(Lang))*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 211 (Revised third ed.), New York: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-95385-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95385-4), [MR](/source/MR_(identifier)) [1878556](https://mathscinet.ams.org/mathscinet-getitem?mr=1878556), [Zbl](/source/Zbl_(identifier)) [0984.00001](https://zbmath.org/?format=complete&q=an:0984.00001)

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