Let $F$ be a field and $p \in F[x] $ an irreducible polynomial. Let $K$ be a finite extension of $F$. How can I prove that if the degree of $p$ doesn't divide $[K:F] $, then $p$ doesn't have any roots in $K$?
2026-04-08 12:40:58.1775652058
Question about degree of a irreducible polynomial and field extensions
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Let $\alpha$ be a root of this irreducible $p \in F[x]$. What is the degree of the extension $F(\alpha) / F$ ?
If $\alpha \in K$, we necessarily have a tower of fields $F \subset F(\alpha) \subset K$. Think about the multiplicativity formula for degrees.