If F is a field and a,b are in F, c is in an extension field, show that F(ac+b)=F(c)

Question

Suppose $F$ is a field such that $a, b \in F$ with $a\neq0$.

If $c$ belongs to some extension $E$ of $F$, prove that $F(c)= F(ac+b)$.

If $c$ is an algebraic element over $F$, this is just easy to prove by looking at $E$ as a vector space over $F$, and then after getting their bases, we can already show the equality of fields above.

However, if $c$ is not specified to be algebraic anymore, I definitely can't use the proof above, and I'm not so sure how to show this.

Can anyone help? Thanks!

Answer 1

$F(c)$ is the smallest field containing both $F$ and $c$.

$F(ac+b)$ is the smallest field containing both $F$ and $ac+b$.

Since $a,b,c \in F(c)$, then $ac+b \in F(c)$. Thus $F(c)$ is a field containing both $F$ and $ac+b$, thus it contains $F(ac+b)$.

In other words, $F(c) \supseteq F(ac+b)$.

In a similar way you prove the opposite inclusion (using $c=[(ac+b)-b]\cdot a^{-1}$).