Problem 3

Determine which of the following polynomials have multiple zeroes over the field $F$ specified. If the polynomial has a multiple zero, state its value.

1. $x^{4}+2x+7$ over $F=\mathbb{Z}_{2}$.

2. $x^{19}+x^{8}+1$ over $F=\mathbb{Z}_{3}$.

3. $2x^{6}+x^{4}+2x^{3}+2$ over $F=\mathbb{Z}_{3}$.

4. $x^{8}+3x^{5}+x^{3}+5$ over $F=\mathbb{Z}_{7}$.