If $\mathbb{F}$ is a field, what is the difference between $\mathbb{F}[a]$ and $\mathbb{F}(a)$? To my understanding $\mathbb{F}(a)$ is the smallest field containing $\mathbb{F}$ and $a$ whereas $\mathbb{F}[a]$ would have to be the polynomials over $\mathbb{F}$? Am I missing something here?
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yes, for a field extension $F\subseteq E$ and an element $a\in E$, the expression $F[a]$ usually denotes the set of all elements of $E$ of the form $\lambda_0+\lambda_1 a+\dots+\lambda_n a^n$ for some $n\in\mathbb{N}$ and $\lambda_i\in F$. equivalently, this is the smallest subring of $E$ containing both $F$ and $a$. exercise: $F[a]$ is itself a field, ie $F[a]=F(a)$, if and only if $a$ is algebraic over $F$. – Atticus Stonestrom Apr 05 '21 at 23:32
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(Intuitive summary, sort of): If $a$ is algebraic over $F$, then $F[a]=F(a)$ because $F[a]$ can be proven to be already a field. If $a$ is transcendental over $F$, then $F[a]$ is a strict subset of $F(a)$. – Apr 05 '21 at 23:33