Question

Q5: (a) Show that p(x) = x^4 + 1 is irreducible over Q. (b) Prove that f(x) = 7x^4 + 19x + 33 is irreducible over Q. (c) Write f(x) = x^3 - x^2 + x - 1 ? Z_3[x] as a product of irreducible polynomials over Z_3.

          Q5: (a) Show that p(x) = x^4 + 1 is irreducible over Q.

(b) Prove that f(x) = 7x^4 + 19x + 33 is irreducible over Q.

(c) Write f(x) = x^3 - x^2 + x - 1 ? Z_3[x] as a product of irreducible polynomials over Z_3.

Added by Mitchell R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shaiju T

06/11/2022

Step 1

Suppose for contradiction that p(x) = f(x)g(x) for some non-constant polynomials f(x), g(x) in Q[x]. Since the degree of p(x) is 4, the degrees of f(x) and g(x) must add up to 4. The only possibilities are (1,3), (2,2), (3,1). However, p(x) has no roots in Q, so Show more…

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Q5: (a) Show that p(x) = x^4 + 1 is irreducible over Q. (b) Prove that f(x) = 7x^2 + 19x + 33 is irreducible over Q. (c) Write f(x) = x^3 - x^2 + x - 1 in Z3[x] as a product of irreducible polynomials over Z3.