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Determine which of the polynomials below is (are)

Chapter 17, Problem 12E

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QUESTION:

Determine which of the polynomials below is (are) irreducible over Q.

a. \(x^5 + 9x^4 + 12x^2 + 6 \)

b. \(x^4 + x + 1 \)

c. \(x^4 + 3x^2 + 3 \)

d. \(x^5 + 5x^2 + 1 \)

e. \((5/2)x^5 + (9/2)x^4 + 15x^3 + (3/7)x^2 + 6x + 3/14\)

Questions & Answers

QUESTION:

Determine which of the polynomials below is (are) irreducible over Q.

a. \(x^5 + 9x^4 + 12x^2 + 6 \)

b. \(x^4 + x + 1 \)

c. \(x^4 + 3x^2 + 3 \)

d. \(x^5 + 5x^2 + 1 \)

e. \((5/2)x^5 + (9/2)x^4 + 15x^3 + (3/7)x^2 + 6x + 3/14\)

ANSWER:

Step 1 of 5

(a)

Eisenstein's criterion (Theorem 17.4) with  implies that  is irreducible over .

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