91Ó°ÊÓ

(i) Prove that \(x\) and \(y\) are relatively prime in \(k[x, y]\), where \(k\) is a field. (ii) Prove that 1 is not a linear combination of \(x\) and \(y\) in \(k[x, y]\).

Prove that if \(R\) is a noetherian ring, then the ring of formal power series \(R[[x]]\) is also a noetherian ring.

(i) Show that \(x^{2}+y^{2}\) is irreducible in \(\mathbb{R}[x, y]\), and conclude that \(\left(x^{2}+y^{2}\right)\) is a prime, hence radical, ideal in \(\mathbb{R}[x, y]\). (ii) Prove that \(\operatorname{Var}\left(x^{2}+y^{2}\right)=\\{(0,0)\\}\). (iii) Prove that Id \(\left(\operatorname{Var}\left(x^{2}+y^{2}\right)\right)>\left(x^{2}+y^{2}\right)\), and conclude that the radical ideal \(\left(x^{2}+y^{2}\right)\) in \(\mathbb{R}[x, y]\) is not of the form \(\operatorname{Id}(V)\) for some algebraic set \(V\). Conclude that the Nullstellensatz may fail in \(k[X]\) if \(k\) is not algebraically closed. (iv) Prove that \(\left(x^{2}+y^{2}\right)=(x+i y) \cap(x-i y)\) in \(\mathbb{C}[x, y]\). (v) Prove that Id \(\left(\operatorname{Var}\left(x^{2}+y^{2}\right)\right)=\left(x^{2}+y^{2}\right)\) in \(\mathbb{C}[x, y]\).

An ideal \(I\) in \(k[X]\) is a monomial ideal if it is generated by monomials: \(I=\) \(\left(X^{\alpha(1)}, \ldots, X^{\alpha(q)}\right)\) (i) Prove that \(f(X) \in I\) if and only if each term of \(f(X)\) is divisible by some \(X^{\alpha(i)}\) (ii) Prove that if \(G=\left[g_{1}, \ldots, g_{m}\right]\) and \(r\) is reduced \(\bmod G\), then \(r\) does not lie in the monomial ideal \(\left(\mathrm{LT}\left(g_{1}\right), \ldots, \mathrm{LT}\left(g_{m}\right)\right.\) ).