Let $a$ and $b$ two elements in an algebraic extension ($K$) of a field $F$. Let $f$ and $g$ the two minimal polynomials of $a$ and $b$.
Show $f$ is irreducible over $F(b)$ if and only if $g$ is irreducible over $F(a)$.
2026-04-09 10:55:08.1775732108
Irreducible polynomials of two algebraic elements of a field
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We have $$ [F(a, b):F] = [F(a, b):F(a)]\cdot [F(a):F] = [F(a, b):F(b)]\cdot [F(b):F] $$ What does $f$ being irreducible over $F(b)$ and $g$ being irreducible over $F(a)$ say about the numbers in the above expression? Can you see why we can't have one without the other?