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I am currently reading notes on Galois Theory and have come upon the following proposition,

Let $f(x)$ be the minimal polynomial of a generator $\alpha$ of a finite field extension $k(\alpha)$ of $k$. Then extension $k(\alpha)/ k$ is normal iff every root $\beta$ of $f(x)=0$ lies in $k(\alpha)$ iff $f(x)$ factors into linear factors in $k(\alpha)[x]$

$\textbf{Question:}$ What is $k(\alpha)/k$ the extension of? Usually I have seen this notation to represent a quotient ring, field, etc. but that doesn't make sense, to me, in this situation. In this case, does the notation represent something else?

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  • $\begingroup$ My guess is an extension of $k$ since it can't be an extension of $k(\alpha)$ but I'm not totally sure. $\endgroup$ – Cameron Williams Jul 18 '15 at 17:30
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The notation $k(\alpha)/k$ is used to say: the field $k(\alpha)$ over $k$. For fields $L/k$ it is similar: $L$ is an extension over $k$. Other notations include $L:k$, $L\mid k$.

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    $\begingroup$ Also $L \supset k$. $\endgroup$ – A.P. Jul 18 '15 at 20:05

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