Abstract Algebra
Let f(x) = x3 ? x ? 1 in Q[x] and let g(x) = x2 + 1. a. Show that f(x) is irreducible in Q[x].
b. Quote general theorems which guarantee that the principal ideal A = (f(x)) is a maximal ideal in Q[x] and the factor ring F = Q[x]/A is a field.
c. Find polynomials r(x) and s(x) such that f(x)r(x) + g(x)s(x) = 1.
Suggestion. Use the same idea as the Euclidean algorithm for Z, invoving divisors and remainders.
d. Explain why the coset g(x) + A is a unit in F and find a coset that represents its multiplicative inverse.
