91Ó°ÊÓ

Prove that a commutative ring \(R\) has a unique one 1 ; that is, if \(e \in R\) satisfies \(e r=r\) for all \(r \in R\), then \(e=1\)

Find all the units in the ring \(\mathbb{Z}[i]\) of Gaussian integers.

(i) Let \(R\) be a domain. Prove that if a polynomial \(f(x) \in R[x]\) is a unit, then \(f(x)\) is a nonzero constant (the converse is true if \(R\) is a field). (ii) Show that \(([2] x+[1])^{2}=[1]\) in \(I_{4}[x]\). Conclude that the statement in part (i) may be false for commutative rings that are not domains.

If \(R\) is a commutative ring, define \(R[[x]]\), the ring of formal power series over \(R\), as the set of all sequences in \(R\). (i) Show that the formulas defining addition and multiplication on \(R[x]\), make sense for \(R[[x]]\), and prove that \(R[[x]]\) is a commutative ring under these operations. (ii) Prove that \(R[x]\) is a subring of \(R[[x]]\). (iii) Denote a formal power series \(\sigma=\left(s_{0}, s_{1}, s_{2}, \ldots, s_{n}, \ldots\right)\) by $$ \sigma=s_{0}+s_{1} x+s_{2} x^{2}+\cdots $$ Prove that if \(\sigma=1+x+x^{2}+\cdots\), then \(\sigma=1 /(1-x)\) is in \(R[[x]]\).

Find the ged of \(x^{2}-x-2\) and \(x^{3}-7 x+6\) in \(\mathrm{F}_{5}[x]\), and express it as a linear combination of them.

Let \(R\) be an arbitrary commutative ring. If \(f(x) \in R[x]\) and if \(a \in R\) is a root of \(f(x)\), i.e., \(f(a)=0\), prove that there is a factorization \(f(x)=(x-a) g(x)\) in \(R[x]\)

Prove the converse of Euclid's lemma. Let \(k\) be a field and let \(f(x) \in k[x]\) be a polynomial of degree \(\geq 1\); if, whenever \(f(x)\) divides a product of two polynomials, it necessarily divides one of the factors, then \(f(x)\) is irreducible.

Let \(k\) be a field, and let \(f(x), g(x) \in k[x]\) be relatively prime. If \(h(x) \in k[x]\), prove that \(f(x) \mid h(x)\) and \(g(x) \mid h(x)\) imply \(f(x) g(x) \mid h(x)\).

(i) If \(k\) is a field, prove that the ring of formal power series \(k[[x]]\) is a PID. (ii) Prove that every nonzero ideal in \(k[[x]]\) is equal to \(\left(x^{n}\right)\) for some \(n \geq 0 .\)