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,

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

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

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

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.,

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

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

Example

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

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

and so, ${\displaystyle \operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a}$.[1] The minimal polynomial of α is X² − 2a X + a² − d b².

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

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

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

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.

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

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]

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

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

• ${\displaystyle \operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L}$
• ${\displaystyle {\text{for any }}\alpha \in K,{\text{ we have }}|\{b\in L\colon \operatorname {Tr} _{L/K}(b)=\alpha \}|=q^{n-1}}$

Theorem.[6] For b ∈ L, let F_b be the map ${\displaystyle a\mapsto \operatorname {Tr} _{L/K}(ba).}$ 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 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 ${\displaystyle \operatorname {GF} (q)=\mathbb {F} _{q}}$ has either 0, 1 or 2 roots in GF(q) (and two roots, counted with multiplicity, in the quadratic extension GF(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 ${\displaystyle x={\sqrt {\frac {c}{a}}}}$ in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:

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

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

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

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

${\displaystyle y=s=\delta +\delta ^{2^{2}}+\delta ^{2^{4}}+\ldots +\delta ^{2^{2m}}}$.

Trace form

When L/K is separable, the trace provides a duality theory 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. An example of where this is used is 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]

See also

• Field norm
• Reduced trace

Notes

1. Rotman 2002, p. 940
2. Rotman 2002, p. 941
3. Roman 1995, p. 151 (1st ed.) harvnb error: no target: CITEREFRoman1995 (help)
4. Lidl & Niederreiter 1997, p.54
5. Mullen & Panario 2013, p. 21
6. Lidl & Niederreiter 1997, p.56
7. Hirschfeld 1979, pp. 3-4
8. Lorenz (2008) p.38
9. 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, 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, 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

Further reading

• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
• Section VI.5 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001