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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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Sandy

Posts: 40
Registered: 7/9/13
Re: Meaning of "irreducible polynomial" in a paper of Mac Lane's
Posted: Oct 29, 2013 12:09 PM
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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
>





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