a) Determine the irreducibility of x20-11 over Q(set of rationals), and use it to prove or disprove that the ideal is a maximal ideal of Q[x].
b) Construct an integral domain R and an element a in R such that a is irreducible but not prime in R.
c) Suppose that R is a principal ideal domain and a in R is irreducible. If a does not divide b in R, prove that a and b are relatively prime.
d) Suppose p in N(set of naturals) is a prime number. Show that every element a in Zp has a p-th root, i.e. there is b in Zp with a=bp.