Why the function field of an algebraic curve $f(x,y)=0$ can be identified with $overlineL(x)[y]/(f)$?

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I am reading the paper An algorithm for computing the Weierstrass normal form. I have a question about the function field of an algebraic curve.



Let $f(x,y)$ be a polynomial in $x,y$. Let $L$ be the field generated by the coefficients of $f$. Let $C$ be the curve given by $f(x,y)=0$.



On page 2, Section 2, it is said that the function field $overlineL(C)$ can be identified with $overlineL(x)[y]/(f)$.



How to prove this fact? Thank you very much.







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    up vote
    1
    down vote

    favorite












    I am reading the paper An algorithm for computing the Weierstrass normal form. I have a question about the function field of an algebraic curve.



    Let $f(x,y)$ be a polynomial in $x,y$. Let $L$ be the field generated by the coefficients of $f$. Let $C$ be the curve given by $f(x,y)=0$.



    On page 2, Section 2, it is said that the function field $overlineL(C)$ can be identified with $overlineL(x)[y]/(f)$.



    How to prove this fact? Thank you very much.







    share|cite|improve this question






















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I am reading the paper An algorithm for computing the Weierstrass normal form. I have a question about the function field of an algebraic curve.



      Let $f(x,y)$ be a polynomial in $x,y$. Let $L$ be the field generated by the coefficients of $f$. Let $C$ be the curve given by $f(x,y)=0$.



      On page 2, Section 2, it is said that the function field $overlineL(C)$ can be identified with $overlineL(x)[y]/(f)$.



      How to prove this fact? Thank you very much.







      share|cite|improve this question












      I am reading the paper An algorithm for computing the Weierstrass normal form. I have a question about the function field of an algebraic curve.



      Let $f(x,y)$ be a polynomial in $x,y$. Let $L$ be the field generated by the coefficients of $f$. Let $C$ be the curve given by $f(x,y)=0$.



      On page 2, Section 2, it is said that the function field $overlineL(C)$ can be identified with $overlineL(x)[y]/(f)$.



      How to prove this fact? Thank you very much.









      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 23 at 8:40









      LJR

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