Field trace

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.

Definition

Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,

m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x

,

is a K-linear transformation of this vector space into itself. The trace, Tr_L/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation.^[1]

For α in L, let σ₁(α), ..., σ_n(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then

\operatorname {Tr} _{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha ).

If L/K is separable 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 and α is in L, then the trace of α is the sum of all the Galois conjugates of α,^[1] i.e.,

\operatorname {Tr} _{L/K}(\alpha )=\sum _{\sigma \in \operatorname {Gal} (L/K)}\sigma (\alpha ),

where Gal(L/K) denotes the Galois group of L/K.

Example

Let $L=\mathbb {Q} ({\sqrt {d}})$ be a quadratic extension of $\mathbb {Q}$ . Then a basis of $L/\mathbb {Q}$ is $\{1,{\sqrt {d}}\}.$ If $\alpha =a+b{\sqrt {d}}$ then the matrix of $m_{\alpha }$ is:

\left[{\begin{matrix}a&bd\\b&a\end{matrix}}\right]

,

and so, $\operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a$ .^[1] The minimal polynomial of α is X² − 2a X + (a² − db²).

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 (a K-linear functional), that is

\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 $\operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .$

Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.

\operatorname {Tr} _{M/K}=\operatorname {Tr} _{L/K}\circ \operatorname {Tr} _{M/L}

.

Finite fields

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

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

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

$\operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L$ .
For any $\alpha \in K$ , there are exactly $q^{n-1}$ elements $b\in L$ with $\operatorname {Tr} _{L/K}(b)=\alpha$ .

Theorem.^[6] For b ∈ L, let F_b be the map $a\mapsto \operatorname {Tr} _{L/K}(ba).$ 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 of L, the trace is called the absolute trace and otherwise it is a relative trace.^[4]

Application

A quadratic equation, ax² + bx + c = 0 with a ≠ 0, and coefficients in the finite field $\operatorname {GF} (q)=\mathbb {F} _{q}$ has either 0, 1 or 2 roots in GF(q) (and two roots, counted with multiplicity, in the quadratic extension GF(q²)). If the characteristic of GF(q) is odd, the discriminant Δ = b² − 4ac indicates the number of roots in GF(q) and the classical quadratic formula gives the roots. However, when GF(q) has even characteristic (i.e., q = 2^h for some positive integer h), these formulas are no longer applicable.

Consider the quadratic equation ax² + bx + c = 0 with coefficients in the finite field GF(2^h).^[7] If b = 0 then this equation has the unique solution $x={\sqrt {\frac {c}{a}}}$ in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:

y^{2}+y+\delta =0,{\text{ where }}\delta ={\frac {ac}{b^{2}}}.

This equation has two solutions in GF(q) if and only if the absolute trace $\operatorname {Tr} _{GF(q)/GF(2)}(\delta )=0.$ 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 $\operatorname {Tr} _{GF(q)/GF(2)}(k)=1.$ Then a solution to the equation is given by:

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

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 via the trace form: the map from L × L to K sending (x, y) to Tr_L/K(xy) is a nondegenerate, 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 in the theory of the different ideal.

The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K.^[8] The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.^[8]

If L/K is an inseparable extension, then the trace form is identically 0.^[9]

Notes

^ ^a ^b ^c Rotman 2002, p. 940
^ Rotman 2002, p. 941
^ Roman 2006, p. 151
^ ^a ^b Lidl & Niederreiter 1997, p.54
^ Mullen & Panario 2013, p. 21
^ Lidl & Niederreiter 1997, p.56
^ Hirschfeld 1979, pp. 3-4
^ ^a ^b Lorenz (2008) p.38
^ Isaacs 1994, p. 369 as footnoted in Rotman 2002, p. 943

References

Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853526-0
Isaacs, I.M. (1994), Algebra, A Graduate Course, Brooks/Cole Publishing
Lidl, Rudolf; Niederreiter, Harald (1997) [1983], Finite Fields, Encyclopedia of Mathematics and its Applications, vol. 20 (Second ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069
Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.
Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6
Roman, Steven (2006), Field theory, Graduate Texts in Mathematics, vol. 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9, Zbl 1172.12001
Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 978-0-13-087868-7