Systematic way to find the quotient field of a ring?

Question

Prove that for the ring $S=F[X,Y]/\left<X^2+Y^2-1\right>$, the quotient field is $F(X)$(${\sqrt {1-X^2}}$).

I think it is possible to prove this by doing long calculation. But is there a systematic way to simplify the problem like by considering $S$ as a ring with one variable and then factor out the relation?

Answer 1

Your equation is not true: on the left $S$ is a dimension 1 ring (or anyway not a field), on the right you have provided a field. (Provided I understand your notation - round brackets are typically used for field extensions. And I am guessing that by $F$ you mean a field.)

I guess you mean to write $F[X][\sqrt{1 - X^2}]$. Adjoining $\sqrt{1 - X^2}$ literally means adjoining a formal variable that squares to $1 - X^2$, which is exactly the construction on the left (that formal variable was called $Y$ instead). You can write down an explicit map if you want.

I guess that this is an answer. What do you think?

Answer 2

Let $F$ be a field of characteristic $\ne2$. From this answer we know that the field of fractions of $F[X,Y]/(X^2+Y^2-1)$ is $F(X)[Y]/(X^2+Y^2-1)$ which can be thought of as $F(X)(\sqrt{X^2-1})$.