Let $k \subset E$ be a field extension, and let $\alpha$ be algebraic over $k$. Let $f$ be the irreducible polynomial of $\alpha$ over $k$, and let $g$ be the irreducible polynomial of $\alpha$ over $E$. Is it true that $g$ always divides $f$?
Asked
Active
Viewed 127 times
1
-
It is true. It follows directly from the fact, that polynomial rings over fields are PIDs. – Severin Schraven Mar 12 '17 at 16:33