Abstract Algebra

Please read it very carefully

Let f(x) = x3 − x −

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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.

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