RANKS OF RELATIVE-UNIT-GROUPS RELATED TO redset(f)

Abstract

Given an irreducible polynomial f in k[X1,…, Xn] (where k is a field) such that k is algebraically closed in the quotient field of A ≔ k[X1,…,Xn]/f k[X1,…,Xn], we show that k(f) is algebraically closed in k(X1,…,Xn). Further, if n ≥ 2 and char k = 0, then we show that the number of k-translates of f that are reducible in k[X1,…, Xn] is bounded above by the rank of U(A)/U(k). Finally, we prove a similar bound for the number of reducible composites of the form Γ(f) with Γ ∈ k[T] monic irreducible.