Determine whether or not each of the following is a field. In each case explain why or why not (i.e. if yes, then explain why each of the axioms is satised and if no, then identify which axiom isn't satised).
(a) The set of positive rational numbers, with the usual multiplication and addition as operation.
(b) The set {a + √5b|a, b ∈ Q, (a, b) ?= (0, 0)} with addition and multiplication as real number as operations.
(c) The set of positive integers with the usual addition and multiplication.
(d) The set of irrational real numbers with multiplication and addition as in R.
(e) The set of 2 × 2 matrices over R with determinant 1 and matrix addition and multiplication as operation.