Show that f(x) = x^4+x^3+3x^2+7x−8 is irreducible over Z.

Expert Solution

We note that is irreducible over if and only if it is irreducible over

And is irreducible over if and only if is irreducible over

That is, is irreducible over

By Eisenstien's criteion for and for we have

(the first terms coefficient), (the coefficients of the rest of the terms) and (coefficient of the constant term)

Therefore, is not reducible over

Which by the above consideration means is irreducible over

Colby Messinger answered 2 years ago

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