Carlitz Extensions

The ring Z has many analogies with the ring Fp[T ], where Fp is a field of prime size p. For example, for nonzero m ∈ Z and nonzero M ∈ Fp[T ], the residue rings Z/(m) and Fp[T ]/M are both finite. The unit groups Z × = {±1} and Fp[T ]× = Fp are both finite. Every nonzero integer can be made positive after multiplication by a suitable unit, and every nonzero polynomial in Fp[T ] can be made monic (leading coefficient 1) after multiplication by a suitable unit. We will examine a deeper analogy: the group (Fp[T ]/M) × can be interpreted as the Galois group of an extension of the field Fp(T ) in a manner similar to the group (Z/(m))× being the Galois group of the mth cyclotomic extension Q(μm) of Q, where μm is the group of mth roots of unity. For each m ≥ 1, the mth roots of unity are the roots of Xm−1 ∈ Z[X], and they form an abelian group under multiplication. We will construct an analogous family of polynomials [M ](X) ∈ Fp[T ][X], parametrized by elements M of Fp[T ] rather than by positive integers, and the roots of each [M ](X) will form an Fp[T ]-module rather than an abelian group (Zmodule). In particular, adjoining the roots of [M ](X) to Fp(T ) will yield a Galois extension of Fp(T ) whose Galois group is isomorphic to (Fp[T ]/M) ×. The polynomials [M ](X) and their roots were first introduced by Carlitz [2, 3] in the 1930s. Since Carlitz gave his papers unassuming names (look at the title of [3]), their relevance was not widely recognized until being rediscovered several decades later (e.g., in work of Lubin–Tate in the 1960s and Drinfeld in the 1970s).