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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: 51 Registered: 7/9/13
Re: Meaning of "irreducible polynomial" in a paper of Mac Lane's
Posted: Oct 29, 2013 12:05 PM

quasi wrote:
> Sandy wrote:
>>
>> 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 non-constant polynomials with coefficients
>> in F.

>
> Right.
>

>> So whether a polynomial is irreducible or not depends on what
>> F is (real field, ring of integers, etc).

>
> Right.
>

>> Mac Lane doesn't say what F is,
>
> Yes, but from the context it's clear that irreducibility is
> intended as irreducibility over the field of complex numbers.
>

>> but there are two fields "in sight", one is k the complex field,
>
> Yes -- that's the one.
>

>> the other is R the real field
>
> But there's no mention of real numbers.
>

>> since curves in the plane (R^2?) are being discussed.
>
> Ah, I see how you got misled.
>
> The intended space for the graph is C^2, not R^2.
>
> True, he called it "the plane", but in this context, he meant
> "the plane C^2" (which is a 2-dimensional vector space over
> the field C).

Thank you. I had considered that it might be any field. The "plane"
then being what I think is called an affine plane. I did not consider
the possibility that it "had to be" k. The fellow might have said so!

> quasi
>

Date Subject Author
10/29/13 Sandy
10/29/13 quasi
10/29/13 Sandy
10/29/13 quasi
10/29/13 Sandy