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Topic: Meaning of "irreducible polynomial" in a paper of Mac Lane's
Replies: 4   Last Post: Oct 29, 2013 12:09 PM

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Posts: 51
Registered: 7/9/13
Meaning of "irreducible polynomial" in a paper of Mac Lane's
Posted: Oct 29, 2013 8:36 AM
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the geometry of an algebraic curve can be reduced to the algebra
of a certain corresponding field; specifically, if k denotes the
field of all complex numbers, and if a curve is defined in the
plane by an *irreducible* polynomial equation f(x,y)=0, then the
corresponding field K is the totality k(x,y) of all rational
functions z = g(x,y)/h(x,y) with complex coefficients... [1]

puzzles me. Specifically, what does "irreducible" mean? I know, or I
think I do, that a polynomial with coefficients in F (some structure) is
said to be irreducible if it cannot be factored into non-constant
polynomials with coefficients in F. So whether a polynomial is
irreducible or not depends on what F is (real field, ring of integers,
etc). Mac Lane doesn't say what F is, but there are two fields "in
sight", one is k the complex filed, the other is R the real field since
curves in the plane (R^2?) are being discussed. But they would make the
polynomials either linear (f = Ax + By + C) or quadratic (f = Ax^2 + Bxy
+ Cy^2 + Dx + Ey + F) respectively, wouldn't they? If Mac Lane intends
that f be either linear or quadratic why not say so? So in what F do
the coefficients of f lie?

[1] Saunders Mac Lane 'Some recent advances in algebra' in
A. A. Albert, ed, 'Studies in modern algebra', MAA, 1963.

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