Powerful Algebra

Algebra Level 3

Let $R$ be a ring.

True or false?

a) If $R$ is a finite field, then $R$ is not algebraically closed.

b) If $p(X) \in R[X]$ then the number of roots of $p(X)$ in $R$ is less or equal than the degree of $p(X)$ .

a)True, b)False a)False b)True a)True, b) True a)False, b) False

1 solution

Guillermo Templado
Jan 4, 2019

a) $X^2 + X + 1$ has no roots in $\mathbb{Z}_2$ , and if $p \ge 3$ , then $X^{p - 1} + 1$ has no roots in $\mathbb{Z}_p$ , (p prime) due to Fermat's theorem.

b) $2X \in \mathbb{Z}_4 [X]$ has 2 roots in $\mathbb{Z}_4$ and its degree is 1.

Powerful Algebra

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