To be or not to be (an ideal)

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In summary, the conversation discusses whether or not the expression x²/y² can be an element of the field F[x,y], as well as the concept of division in the ring F[x,y]. It is mentioned that x²/y² is undefined in this context, but can be considered in the field F(x,y). The origin of the question is discussed, which involves proving F[x,y]/(x²/y²) as a vector space, and the misunderstanding that occurred regarding the notation F(x,y)/(x²/y²). The conversation concludes by clarifying that the expression should actually be F[x,y]/(x², y²), which makes more sense in the context of ideals in a field.
  • #1
matness
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let F be a field.x^2/y^2 is not an element of F[x,y](is it?)
(x^2/y^2) can or can not be ideal in F[x,y] ?
 
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  • #2
Strictly speaking, the expression

x²/y²

is undefined, because one cannot divide in the ring F[x, y]. And because x²/y² is undefined, so is the expression (x²/y²).


We're usually more generous with notation, though; rather than leave x²/y² undefined, we implicitly shift our attention to the field F(x, y), which does contain an element by that name.

Where did this come from?
 
  • #3
the origin of my question:i have to prove F[x,y]/(x^2/y^2) is a vector space it seemed a bit meaningless anddid not remember fraction fields
probably it was F(x,y)/(x^2/y^2) and i misread it
sorry:uhh:
 
  • #4
F(x, y) / (x² / y²) doesn't make much sense either; the only ideals of a field are the zero ideal and the whole field itself.

It probably said F[x, y] / (x², y²)
 
  • #5
now it is clear thank you very much
 

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