User:Michellepoliseno /Math735 HW8

9.2.3 Let $ f(x) \ $ be a polynomial in $ F[x] \ $. Prove that $ F[x]/(f(x)) \ $ is a field if and only if $ F(x) \ $ is irreducible. [Use Proposition 7, Section 8.2]

Assume that $ F \ $ is a field. Then $ F[x] \ $ is a Euclidean Domain, which implies that $ F[x] \ $is a Principal Ideal Domain. Then by proposition 7 in section 8.2 implies that prime ideals in $ F[x] \ $ are maximal. Proposition 14 from Section 7.4, in a commutative ring $ R \ $, all maximal ideals are prime. Thus, (*) in $ F[x] \ $ an ideal is prime $ \iff \ $ it is a maximal ideal.

$ F[x]/(f(x)) \ $ is a field $ \iff (f(x)) \ $ is a max ideal of $ F[x] \ $, by Proposition 12 from section 7.4. $ \iff (f(x)) \ $ is a prime ideal of $ F[x] \ $, by (*) $ \iff f(x) \ $ is prime in $ F[x] \ $, by Definition (2) on page 284 of the text. $ \iff f(x) \ $ is irreducible in $ F[x] \ $, by Proposition 11 in Section 8.3.

9.3.2 Prove that if $ f(x) \ $ and $ g(x) \ $ are polynomials with rational coefficients whose product $ f(x)g(x) \ $ has integer coefficients, then the product of any coefficients of $ g(x) \ $ with any coefficient of $ f(x) \ $ is an integer.