Exercise 7.2.1 (Irreducible over rationals)

Question

Try to find an example of an irreducible polynomial
of degree $d$ with fewer than $d$ distinct roots in its splitting field.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
An irreducible polynomial over a field of characteristic 0 has distinct roots in its splitting field.
Therefore we must consider field of characteristic $p >0$, where $p$ is prime. Hence if we have the field extension $\mathbb{F}_p(t):\mathbb{F}_p(t^p)$,
and we consdier $t \in \mathbb{F}_p(t)$ its minimal polynomial over $\mathbb{F}_p(t^p)$ is $X^p -t^p = (X-t)^p$. We get to the last step from the Frobenius automorphism.

Exercise 7.2.1 (Irreducible over rationals)

Answers

Comments

Exercise 7.2.1 (Irreducible over rationals)

Answers

Comments

Add answer