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I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105):

In fact, what I didn't understand is the following notation in the proof of this proposition:

Why $k[X,Y]/(F)=k[x,y]$ with small letters? What's the difference between $k[X,Y]$ and $k[x,y]$ in this context?


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up vote 5 down vote accepted

$k[x,y]$ just refers to a $k$-algebra generated by $2$ elements $x, y$, which need not be algebraically independent over $k$ (as $X, Y$ are). Indeed, $x, y$ are the images $X + (F), Y + (F)$ in the quotient ring $k[X,Y]/(F)$.

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Thank you for your answer, it helped a lot! – user85493 Aug 19 '14 at 7:44
Just one question, what do you mean by $X, Y$ are algebraically independent over $k$? $X$ and $Y$ are just indeterminates, not elements. – user85493 Aug 19 '14 at 7:46
@user85493: Indeterminates are, in fact, elements in the polynomial ring, and are algebraically independent (essentially by definition, which is that they satisfy no polynomial with coefficients in $k$). In contrast, note that in this example, if $F \ne 0$, then $x, y$ are not algebraically independent – zcn Aug 19 '14 at 9:02

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