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When we talk about fields, what is the difference between $F(a)$ and $F[a]$? I have usually understood that $F[a]$ stands for polynomials over the field $F$ whereas $F(a)$ stands for $F(a)=\{c_{1}+c_{2}\cdot a: c_1,c_2 \in F\}$? Am I correct here or is there anything else that I should take into consideration as well?

Bill Dubuque
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  • ones a polynomial ring, the other is field of fractions – ureui Mar 20 '21 at 08:14
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    You're definition of $F(a)$ is off. $F(a)$ is defined as the smallest field containing $F$ and $a$. This means, for example, that $\mathbb{Q}(\sqrt2) = \mathbb{Q}$ while if $X$ is just a formal variable $F(X)$ is the rational functions. More concretely, $F[a]$ gives you a ring while $F(a)$ is the field of fractions of that ring. – memerson Mar 20 '21 at 08:20
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    @memerson: you have written $\mathbb{Q}(\sqrt{2})=\mathbb{Q}$. Did you perhaps mean $\mathbb{Q}(\sqrt{2})=\mathbb{Q}[\sqrt{2}]$? – ancient mathematician Mar 20 '21 at 08:58
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    @ancientmathematician, I meant neither. I actually meant $\mathbb{Q}(2)$ (for which the equation holds), but I put the square root in there by mistake. Although $\mathbb Q$ and $\sqrt 2$ are a great example of when $F(a) = F[a]$ – memerson Mar 20 '21 at 09:47

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