quasi wrote: > Sandy wrote: >> >> ... 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? > > As I mentioned in my previous reply, the intended field of > coefficients is C, not R, but even so, it's not true that > irreducible multivariate polynomials with real coefficients > must be linear or quadratic.
That was my big mistake (I have decided I can be forgiven for not knowing that "plane" meant k^2!), I assumed that irreducible multivariate polynomials must be linear or quadratic (or just linear over C). I am an idiot. Sorry.
> For example the polynomial x^3 + y^3 + 1 is irreducible in the > ring R[x,y]. In fact, it's even irreducible in the ring C[x,y]. > > What is true is that if a nonconstant polynomial f in R[x] is > irreducible in R[x], then deg(f) = 1 or 2. > > Equivalently, if a nonconstant _homogeneous_ polynomial g in > R[x,y] is irreducible in R[x,y], then deg(g) = 1 or 2. > > quasi >