Normal Elements in Finite Fields

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If u := g∈G agg and v := g∈G bgg, then α = uβ and β = vα. Thus uvα = α and vuβ = β. Since αPand β are normal elementsP this implies that uv =1= vu, hence u is a unit in K[G]×. Therefore every normal element β is a K[G]× multiple of α. A weaker version of Theorem 2 appears implicitly in Suwa [11, Cor. 1.7] where it is traced back to an argument of Serre [10, Chp. IV, Prop. 7]. Their statement is an equivalence between the existence of a normal basis of a Galois algebra and of a certain pull-back diagram involving units in a group scheme associated to the Galois group ring. Lemma 3 below is the function ﬁeld analog of the classic formula for Euler’s totient function 1 ϕ(n)= n 1 − , p Yp|n where the product on the right is taken over all prime divisors of n without multiplicity. Note that ϕ(n) may be deﬁned as the number of multiplicative units modulo n. Lemma 3 is well-known, see [9, Prop. 1.7] for example. We give a proof for completeness.

Lemma 3. If f(x) ∈ Fq[x] is non-constant, then

× deg(f) 1 |(Fq[x]/(f)) | = q 1 − , qdeg(p) p(xY)|f(x) where the product is taken over all monic irreducible factors of f(x) without multiplicity. × Proof. Let ϕ(f) := |(Fq[x]/(f)) |. Then the probability of a uniformly random element of deg(f) × Fq[x]/(f) being a unit is, on one hand, simply ϕ(f)/q . On the other hand, u(x) ∈ (Fq[x]/(f)) is equivalent to u(x) not being divisible by any irreducible p(x) dividing f(x), and these events are independent by the Chinese Remainder Theorem. Hence ϕ(f) 1 = 1 − . qdeg(f) qdeg(p) p(xY)|f(x)

Proof of Theorem 1. The Galois group of Fqn /Fq is cyclic of order n generated by the Frobenius q auotomorphism σ : a 7→ a . Therefore the group algebra Fq[hσi] is naturally isomorphic to n Fq[x]/(x − 1) by σ 7→ x. Theorem 2 implies that the number of normal elements Nn(q) is equal to the number of units in Fq[hσi], hence by Lemma 3

n × n 1 Nn(q)= |(Fq[x]/(x − 1)) | = q 1 − . (1) qdeg(p) p(x)Y|xn−1 m n d pm If n = dp where p does not divide d, then x − 1 = (x − 1) in Fq[x], hence the product (1) may be taken over irreducibles p(x) | xd −1. By Galois theory these irreducible factors correspond to orbits of Frobenius on the dth roots of unity. The orbit of a primitive eth root of unity has length oe(q), the multiplicative order of q modulo e, and there are ϕ(e)/oe(q) such orbits. Hence ϕ(e)/oe(q) n 1 Nn(q)= q 1 − . qoe(q) Ye|d The proofs of Theorem 1 in [1, 3, 5, 7] use additive polynomials to count normal elements in Fqn /Fq. Recall that a polynomial f(x) ∈ Fq[x] is additive if f(x + y) = f(x)+ f(y) or d qi equivalently if f(x)= i=0 aix . Additive polynomials are an essentially positive characteristic P NORMAL ELEMENTS IN FINITE FIELDS 3 phenomenon and thus this approach gives the impression that the enumeration in Theorem 1 hinges on some special feature of positive characteristic fields. However, the ring of additive polynomials is isomorphic to the Galois group ring Fq[hσi], and the latter generalizes to Galois extensions in any characteristic. The important underlying structure is the free transitive action of the units in the Galois group ring on normal elements (Theorem 2) and the structure of finite fields only comes n × in to count (Fq[x]/(x − 1)) . For example, if L/K is any degree n cyclic Galois extension, then each choice of normal element in L/K and generator of the Galois group provides an explicit bijection between all normal elements and units in the algebra K[x]/(xn − 1), a product of cyclotomic extensions of K. Ore [7, Chp. 1] proves Theorem 1 by studying the minimal additive polynomial associated to each element of Fqn and observing that normal elements are precisely those whose minimal n additive polynomial is xq − x. He then uses an inclusion-exclusion argument to arrive at the product formula (1). Akbik [1] independently came to essentially the same proof nearly 60 years later. Lenstra and Schoof [5, Sec. 1] give an exposition of Ore’s proof in their work on primitive normal bases; the ideas behind our proof appear there between the lines. They describe Ore’s results as pertaining to the Galois module structure of Fqn /Fq, but neither explicitly state that the ring of additive polynomials is the group ring of Gal(Fqn /Fq) nor refer to Theorem 2 directly. They do allude to the general connection between normal elements and units in the Galois group ring in their assertion [5, (1.15) Pg. 221]. Perlis [8, Lem. 1] gives a criterion for an element in Fqn to generate an normal basis which is equivalent to Ore’s characterization in terms of minimal additive polynomials. Gao [2, Pg. 18] notes that Perlis’s result can be used to count the number normal elements. Perlis [8, Thm. 1] also shows that normal elements may be detected by the non-vanishing of their trace, and Gao [2, Cor. 2.4.7] uses this to give another enumeration of normal elements. Acknowledgments. We thank Andrew O’Desky for sharing Theorem 2 and clarifying its relation to the work of Suwa and Serre. We thank Julian Rosen and Mike Zieve for bringing references to our attention, in particular [2, 5, 7, 8]. We thank Bob Lutz and Andrew O’Desky for comments on an earlier draft. Finally we are grateful to Jeff Lagarias for helpful feedback and encouragement.

REFERENCES [1] S. Akbik, Normal generators of finite fields, J. Number Theory, 41, (1992), 146-149. [2] S. Gao, Normal bases over finite field, Dissertation, University of Waterloo, (1993). [3] K. Hensel, Über die Darstellun der Zahlen eines Gattungsbereiches für einen beliebigen Primdivisor, Journal für Mathematik, 103, (1888), 230-237. [4] S. Lang, Algebra, 211, Springer Science & Business Media, (2002). [5] H. W. Lenstra, R. Schoof, Primitive normal bases for finite fields, Math. Comp., 48, No. 177, (1987), 217-231. [6] A. O’Desky, J. Rosen, Group rings, normal bases, and étale algebras, in preparation. [7] O. Ore, Contributions to the theory of finite fields, Trans. Amer. Math. Soc., 36, No. 2, (1934), 243-274. [8] S. Perlis, Normal bases of cyclic fields of prime-power degree, Duke Math. J. 9, No. 3, (1942), 507-517. [9] M. Rosen, Number Theory in Function Fields, 210, Springer Science & Business Media, (2002). [10] J.-P. Serre, Algebraic groups and class fields, 117, Springer Science & Business Media, (2012). [11] N. Suwa, Around Kummer theories, RIMS Kôkyûroku Bessatu B12, (2009), 115-148.

DEPT. OF MATHEMATICS,UNIVERSITY OF MICHIGAN,ANN ARBOR, MI 48109-1043, E-mail address: [email protected]