1. RANKS OF RELATIVE-UNIT-GROUPS RELATED TO redset(f).
- Author
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OGLE, JACOB and Abhyankar, S.
- Subjects
- *
RANKING (Statistics) , *GROUP theory , *SET theory , *POLYNOMIALS , *ALGEBRAIC fields , *PROOF theory , *FACTORIZATION - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2012
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