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1. Introduction. For a prime p, the well-known Wilson congruence says that (p − 1)! ≡ −1 modulo p. A prime p is called a Wilson prime if the congruence above holds modulo p 2 . We now quote from [R96, pp. 346, 350]: 'It is not known whether there are infinitely many Wilson primes. In this respect, Vandiver wrote: This question seems to be of such a character that if I should come to life any time after my death and some mathematician were to tell me it had been definitely settled, I think I would immediately drop dead again.' Ribenboim also mentions that search (by Crandall, Dilcher, Pomerance [CDP97]) up to 5 • 10 8 produced the only known Wilson primes, namely 5, 13, and 563, as discovered by Goldberg in 1953 (one of the first successful computer searches involving very large numbers). See [R96, Dic19] for other historical references.Many strong analogies [Gos96,Ro02,Tha04] between number fields and function fields over finite fields have been used to benefit the study of both. These analogies are even stronger in the base case Q, Z ↔ F (t), F [t], where F is a finite field. We study the concept of Wilson prime in this function field context, and in contrast to the Z case, we exhibit infinitely many of them, at least for many F . For example, ℘ = t 3 * 13 n − t 13 n − 1 are Wilson primes for F 3 [t].We also show strong connections between Wilson's and Fermat's quotients, and also between refined Wilson residues and discriminants. Moreover, we introduce analogs of Bell numbers in the F [t] setting.2. Wilson primes. Let us fix some basic notation. We use the standard conventions that empty sums are zero and that empty products are one.

Abstract. Let q be a prime power and let Fq be the finite field with q elements. For each polynomial Q(T ) in Fq[T ], one could use the Carlitz module to construct an abelian extension of Fq(T ), called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of Fq(T ), similar to the role played by cyclotomic number fields for abelian extensions of Q. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in Fq [T ]. Two types of properties are obtained for the l-parts of the class numbers of the fields in this tower, for a fixed prime number l. One gives congruence relations between the l-parts of these class numbers. The other gives lower bound for the l-parts of these class numbers.