Topics in algebra Chapter IV: Commutative rings and modules I - 1

Theorem 2.39 (Eisenstein’s Criterion) Let D be a U.F.D. with quotient
field F . If f(x) =
n
aix i ∈ D[x], deg(f) ≥ 1 and p is an irreducible
i=0
element of D such that
p ∤ an; p|ai for i = 0, . . . , n − 1; p 2 ∤ a0,
then f is irreducible in F [x]. If f is primitive then f is irreducible in D[x].
Proof. Since C(f) is a unit in F , we may assume that f is primitive. Suppose
that f is reducible in F [x]. Then f is reducible in D[x]. Therefore there are
m
primitive polynomials g = bix i k
, h = cix i of D[x] of positive degree
i=0
such that f = gh. Since p|a0 and p 2 ∤ a0, we may assume that p|b0 and p ∤ c0.
Let t be the integer such that
i=0
p|bi for i = 0, . . . , bt; p ∤ bt+1.
Notice that t + 1 ≤ m < n as k ≥ 1. Therefore, since at+1 = bt+1c0 + btc1 +
· · · + b0ct+1 is a multiple of p, we conclude that bt+1c0 is a multiple of p, a
contradiction. Thus, f is irreducible in F [x].
Problem set
1. Determine all prime ideals and maximal ideals of Zn.
2. Let f : R −→ S be a ring homomorphism and Q is a prime ideal of S;
then f −1 (Q) is a prime ideal of R.
3. Let f : R −→ S be a ring epimorphism and Q is a maximal ideal of
S; then f −1 (Q) is a maximal ideal of R.
4. Let N be the ideal of all nilpotent elements of R; then R/N has no
nontrivial nilpotent element.
5. Find an integral domain R and an element a ∈ R such that a is irreducible
but a is not a prime element of R.
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