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.