Advanced Algebra II
Homework 12 due on Jun. 4, 2004
(1) Let k be a field. Consider the inclusion A := k[x2− 1] ⊂ B :=
k[x]. It’s clearly an integral extension. Let n = (x − 1) be a maximal ideal of B. And let m := n ∩ A. Describe m. Is Bn integral over Am?
(2) Let A be an integrally closed domain and K be its quotient field.
Let f (x) ∈ A[x] be a monic polynomial. If f (x) is reducible in K[x] then it’s reducible in A[x].
(Hint: consider roots of f (x).)
(3) Let G be a finite subgroup of automorphism of a ring A. Let AG denote the subring of G-invariants, i.e.
AG := {x ∈ A|σ(x) = x, ∀σ ∈ G}.
Show that A is integral over AG.
(4) Let A ⊂ B be a finitely generated integral extension. Let p ∈ SpecA. Show that there are only finitely many prime ideals lying over p.
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