A question regarding reason behind assumptions in 2 proofs of Cyclotomic extensions

Question

I have self studied section Cyclotomic extensions( page 298 ) from Algebra by Thomas Hungerford and I have 1 question

Consider these 2 theorems :

Question: why does in both the theorems there is an assumption that char K doesn't divides n?

I have read both the proofs but couldn't find the reasoning behind it.

Can you please tell why it happens?

Answer 1

For separability reasons. If the characteristic of $K$ is $p$ which divides $n$, then $$X^n-1=X^{kp}-1=(X^k-1)^p,$$ for some $k$, so the $n$-th roots of unity are actually $k$-th roots of unity. In particular if $k=1$, then there are no primitive roots of unity.