Show that the polynomial
3x^(5) + 15x⁴+20x³ + 10x + 20 is irreducible over Q. Is the polynomial 3x² + x + 4 irreducible over Z7 ? Give
reasons.
Let us show that the polynomial is irreducible over .
We use Eisenstein's criterion. Suppose we have the following polynomial with integer coefficient:
If there exists a prime number p such that the following three conditions all apply:
then is irreducible over the rational numbers.
In our case, let .
Then 5 | 15, 5 | 20, 5 | 10, 5 | 20.
Also 5 does not divide 3, and
25 does not divide 20.
Therefore, according to Eisenstein's criterion, is irreducible over .
Is the polynomial irreducible over ?
Since , we conclude that this polynomial is reducible over (note that 8 = 1 in the field ).
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