Fields3

From Exampleproblems

Prove that the polynomial is irreducible: $x^4+4x^3+6x^2+2x+1\isin\mathbb{Z}[x]\,$

The image of $f(x)\,$ in $\mathbb{Z}/3\mathbb{Z}[x]\,$ is $f'(x)=x^4+x^3+2x+1\,$ .

There are no linear factors: $f'(0)=1, f'(1)=2, f'(2)=2\,$ .

Checking division by all quadratic irreducibles will show that $f'(x)\,$ is irreducible in $\mathbb{Z}/3\mathbb{Z}[x]\,$ and since $f(x)\,$ is primitive, $f(x)\,$ is irreducible over $\mathbb{Z}\,$ .

Main Page : Abstract Algebra : Fields

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