Let i be a monomial ideal in kt1tq and x vi aq k the

Question

Question: Let I be a monomial ideal in K[t₁,...,t_q] and X = V(I) ⊂ Aq K the associated monomial variety. If K is infinite, prove that X is irreducible if and only if X = V (ti₁ ,...,t_ir )? 0 Let K be an infinite field. Prove:

(a) Aq K is an irreducible variety,

(b) any two non-empty open sets of Aq K intersect,

(c) any non-empty open set of AqK is dense,

(d) (K∗)q is an open set of AqK,

(e) (K∗)q is not an affine variety.