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Given that $\mathbb{Q}[\alpha]=\{g(\alpha), g\in\mathbb{Q}[X]\}, \alpha\in\mathbb{C}$ we consider the field extension $K(X)/K$ for an arbitrary field $K$ and $K(X)$ denoting the quotientfield of $K$ in $X$.

Since $X$ is a variable, I don't understand the meaning of the notation $K(X)/K$. In my book $K(X)$ is only defined, if $X$ would be a complex number.

Also, the author of the book defines a field extension as a pair $K\subset L$ with $K,L$ being fields, and thats it.

Does this notation even make sense if I don't modify the definition of a field extension?

EpsilonDelta
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1 Answers1

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I guess you know the ring of polynomials $K[X]$ over a field $K$? Since this is an integral domain you can form the quotient field of that ring, which then is the so called field of rational functions $K(X)$ over $K$, i.e.$$ K(X)= \{\frac{f}{g} | f,g\in K[X], g\neq 0\}. $$ Since both are fields and $K \subset K(X)$ in the obvious manner, $K(X)$ is a field extension of $K$.

Did you define in your course/book what $K(\alpha)$ and what $K[\alpha]$ is and when they are equal (and when they are not)? Maybe that's where your confusion comes from.

Verdruss
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