91Ó°ÊÓ

Let \(f(x)=x^{n-1}+x^{n-2}+\ldots+x+1 \in \mathbb{Q}[x],\) where \(n\) is not prime. Show that \(f(x)\) is not irreducible over \(\mathrm{Q}\).

For any ring \(R\) define the set \(R[[x]]\) of formal power series in the indeterminate \(x\) with coefficients from \(R\) to be the set of all infinite formal sums $$ \Sigma_{i=0}^{\infty} a_{i} x^{i} $$ with all \(a_{i}\) in \(R\). $$ \text { Find the multiplicative inverse of } x+1 \text { in } \mathbb{Z}[[x]] \text { . } $$

Let \(F\) be a field and let \(I=\langle f(x)\rangle\) and \(J=\langle g(x)\rangle\) be two ideals in \(F[x]\). Prove a general result about the generator of the ideal \(I \cap J\).

Let \(F\) be a field \(f(x)\) and \(g(x)\) in \(F[x]\) be monic polynomials. Show that if \(g(x)\) is irreducible over \(F\) and \(f(x) \mid g(x),\) then either \(f(x)=1\) or \(f(x)=g(x)\).

Let \(F\) be a field and \(F(x)\) the field of rational functions over \(F\). (See Definition 8.1.20.) Then \(f(x) / g(x) \in F(x)\) is said to be in reduced form if \(g(x)\) is monic and \(\operatorname{gcd}(f(x), g(x))=1\). Show that every element \(f(x) / g(x) \in F(x)\) can be put in reduced form in a unique way.

Find the multiplicative inverse of the indicated element in the indicated field. $$ x^{2}+1 \quad \text { in } \mathbb{Z}_{3}[x] /\left\langle x^{3}+2 x+1\right\rangle $$

Find the multiplicative inverse of the indicated element in the indicated field. $$ x^{2}-x+1 \quad \text { in } \mathrm{Q}[x] /\left\langle x^{3}-2\right) $$

Find the multiplicative inverse of the indicated element in the indicated field. $$ x+3 \quad \text { in } Q[x] /\left\langle x^{2}-2\right\rangle $$