Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$.
Attempt: Since $f(x),g(x)$ are irreducible over $F \implies a,b \notin F$.
$f(x)$ is irreducible over $F(b) $ and $f(x)$ is irreducible over $F \implies a \neq b$ (As, $a$ is the zero of $f(x)$)
Which means $b \notin F(a)$ either $\implies g(x)$ is irreducible over $F(a)$.
Similarly, the other half can be proved in a similar way.
Is my solution attempt correct?
Thank you for your help..
Best Answer
Hint $\qquad \begin{array}{ccc} & F(\alpha,\beta)\ &\\ \color{#c00}x\nearrow\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \nwarrow \color{#0a0}y\\ F(\alpha)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! & &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! F(\beta)\\ & a\nwarrow\qquad\nearrow b \\ & F & \end{array} \Rightarrow\ \ \ {xa = yb}\ \ \ \Rightarrow\!\!\!\!\!\! \overset{\Large \stackrel{g\ {\rm irred\ over\ } F(\alpha)\ \ \ \ \ }\Updownarrow}{\color{#c00}{x=b}}\!\!\!\!\!\!\!\!\!\iff\!\!\!\!\!\!\!\!\! \overset{\Large \stackrel{\ \ f\ {\rm irred\ over\ }F(\beta)}\Updownarrow_\phantom{I^{I^I}}\!\!\!\!\!\!\!\!}{\color{#0a0}{y = a}}$