Abstract Algebra
Please read it very carefully
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].
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Abstract Algebra
Please read it very carefully
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. Quot
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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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