Sandy
Posts:
40
Registered:
7/9/13


Meaning of "irreducible polynomial" in a paper of Mac Lane's
Posted:
Oct 29, 2013 8:36 AM


This:
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 nonconstant 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.

