Lerch’s Theorems over Function Fields

In this work, we state and prove Lerch’s theorems for Fermat and Euler quotients over function fields defined analogously to the number fields. 1. Results The Fermat’s little theorem states that if p is a prime and a is an integer not divisible by p, then ap−1 ≡ 1 mod p. This gives rise to the definition of the Fermat quotient of p with base a, q(a, p) = ap−1 − 1 p , which is an integer. This quotient has been widely investigated and applied by many authors (see, e.g., [1, 2, 3, 7]). In 1905, Lerch [4] introduced and studied a generalization of the Fermat quotient for an arbitrary composite modulus m ≥ 2 based on Euler’s theorem, so called the Euler quotient. The following congruence is due to Lerch [1, 4]: Theorem 1. [Lerch, 1905] If a and m ≥ 2 are relatively prime integers, then q(a,m) = aφ(m) − 1 m ≡ m ∑ r=1 gcd(r,m)=1 1 ar [ar m ] mod m, where [x] denotes the greatest integer ≤ x. It is well-known that the ring of integers Z has many properties in common with A = Fq[x], the ring of polynomials over the finite field Fq in an indeterminate x. Over a function field, we have not only the result parallel to Fermat’s little theorem, but we also have Euler’s theorem on A (see, Chapters 1 and 3 of [5]). INTEGERS: 10 (2010) 26 Let Fq be a finite field with q elements and set A = Fq[x]. Let a ∈ A and P be irreducible over A. Write |P | for qdeg P . If P does not divide a, we know that a|P |−1 ≡ 1 mod P , which is analogous to Fermat’s little theorem. Fix d | q − 1. For P not dividing a, let ( a P )d be the unique element of F × q such that a |P |−1 d ≡ ( a P )d mod P. If P | a, we let ( a P )d = 0. The symbol ( a P )d is called the d-th power residue symbol. We define thus the polynomial qd(a, P ) = a |P |−1 d − ( a P )d P , called the Fermat quotient of degree d for P with base a. For d = 1, aq1(a, P ) = a|P | − a P is the Fermat quotient studied in [6] by Sauerberg and Shu. Another extension of the Fermat quotient, called the Euler quotient, is defined from Euler’s theorem as follows: For a and f polynomials in A with gcd(a, f) = 1, one has a result parallel to Euler’s theorem, namely, a ≡ 1 mod f, where Φ(f) denotes the cardinality of the unit group (A/fA)×. Following Lerch [4] and Agoh et al. [1], the Euler quotient for f with base a is given by the polynomial q(a, f) = aΦ(f) − 1 f . Observe that Φ(P ) = |P | − 1 if P is irreducible. Hence the Euler quotient is a generalization of the Fermat quotient q1(a, P ). In this work, we study function field analogs of Lerch’s theorem for Euler and Fermat quotients. We present our versions of Lerch’s congruence for Euler and Fermat quotients in Theorems 2 and 3, respectively. Theorem 2. For polynomials a and f in A with gcd(a, f) = 1, we have