3. Let F be a field. Prove that the set R of polynomials in F(a] whose coefficient of x is 0 is a subring of Fa] and that R is not a U.F.D. (hint: show that a = (a²)³ = (x³)² gives two distinct factorizations of a® into irreducibles.) %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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3. Let F be a field. Prove that the set R of polynomials in F[x] whose coefficient of x is 0 is
a subring of F[æ] and that R is not a U.F.D. (hint: show that x = (x²)³ = (x³)² gives two
distinct factorizations of 2® into irreducibles.)
%3D
Transcribed Image Text:3. Let F be a field. Prove that the set R of polynomials in F[x] whose coefficient of x is 0 is a subring of F[æ] and that R is not a U.F.D. (hint: show that x = (x²)³ = (x³)² gives two distinct factorizations of 2® into irreducibles.) %3D
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