Why maximal ideals in a ring of two-variable polynomials with complex coefficients are those generated by x - c, and y - d, for some complex c and d?

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- February 9th 2009, 10:45 PMDifferentMaximal ideals of C[x,y]
Why maximal ideals in a ring of two-variable polynomials with complex coefficients are those generated by x - c, and y - d, for some complex c and d?

Thanks - February 10th 2009, 12:58 AMNonCommAlg
this is a special case of weak Nullstellensatz

*****. one side is trivial because the other side is much deeper: let be a maximal ideal of then is clearly a

finitely generated algebra, which is also a field. a well-known result in commutative algebra says that a finitely generated domain over a field F is a field iff it's algebraic over F******. so

must be an algebraic extension of which is possible only if because is algebraically closed. now let be the image of under the natural projection

then clearly and thus but we already proved that is a maximal ideal of therefore

*****if is any algebraically closed field, then maximal ideals of polynomial ring are exactly the ideals

******see for example page 162 of the book "Graduate Algebra: Commutative View". the author is Louis Halle Rowen. - February 10th 2009, 04:33 AMDifferent
Thanks a lot. Funny enough, I understand the other direction, but could you explain why C[x,y] / <x - c, y - d> is isomorphic to C?

I assume the isomorphism sends x to c and y to d....so, why does a polynomial f(x,y) s.t. f(c,d)=0 belongs to the ideal <x - c, y - d> ? - February 10th 2009, 02:30 PMNonCommAlg
- February 10th 2009, 04:49 PMDifferent
All clear now. Thanks!