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Introduction to Abstract Algebra I MATH 321 Fall 2025

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Worksheet Homework 2

1.

Explain how you know each of the following polynomials are irreducible in the given ring. Reference the appropriate theorem when appropriate.
  1. \(f(x) = x^3 + 3x^2 + 4x + 5\) in \(\Q[x]\text{.}\)
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  2. \(g(x) = x^2 + 3x + 5\) in \(\Z_7[x]\text{.}\)
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  3. \(h(x) = x^5 + 6x^4 + 9x^2 + 3\) in \(\Q[x]\text{.}\)
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2.

You cannot apply Eisenstein’s criterion to the polynomial \(p(x) = x^3 - 3x + 1\text{.}\) However, we can apply it to \(p(x+c)\) for a carefully picked value of \(c\text{.}\)
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(a)

Prove that for any polynomial \(p(x)\) and any constant \(c\text{,}\) if \(p(x+c)\) is irreducible, then so is \(p(x)\text{.}\) (Hint: prove the contrapositive.)
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(b)

For \(p(x) = x^3 - 3x + 1\text{,}\) prove that \(p(x+2)\) is irreducible.
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3.

Factor the polynomial \(p(x) = x^7 + 2x^6 - 3x - 6\) completely (into irreducible factors) over \(\Q\text{,}\) then over \(\bC\text{,}\) and then over \(\R\text{.}\) (Hint: do the factoring in that order.)
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4.

True or false: \(x^4 + 20x^3 + 5x^2+ 10x + 15\) is irreducible in \(\R[x]\text{.}\) Briefly explain.
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