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Thread: Algebraic geometry---> irreducible polynomial

  1. #1
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    Algebraic geometry---> irreducible polynomial

    Hello, I am trying to show that y^5 -x^2 is irreducible in R[x,y]. (R represents the set of real numbers and R[x,y] , the polynomial ring in two variables)

    I have tried everything but still don't know how to show this!

    Here's what I tried:

    y^5 -x^2 = fg , where f&g are non constant polynomials in R[x,y].

    Then let x=t^5 and y=t^2.

    so, 0=f(t^5,t^2)g(t^5,t^2), which implies

    either f(t^5,t^2)=0 or......

    My professor said this is the best approach and I should find a contradiction.

    Any help will be grateful thanks.
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  2. #2
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    Quote Originally Posted by Algebraicgeometry421 View Post
    Hello, I am trying to show that y^5 -x^2 is irreducible in R[x,y]. (R represents the set of real numbers and R[x,y] , the polynomial ring in two variables)

    I have tried everything but still don't know how to show this!

    Here's what I tried:

    y^5 -x^2 = fg , where f&g are non constant polynomials in R[x,y].

    Then let x=t^5 and y=t^2.

    so, 0=f(t^5,t^2)g(t^5,t^2), which implies

    either f(t^5,t^2)=0 or......

    My professor said this is the best approach and I should find a contradiction.

    Any help will be grateful thanks.
    look at $\displaystyle -x^2 + y^5$ as an element of $\displaystyle D[x],$ where $\displaystyle D=\mathbb{R}[y].$ note that the units of $\displaystyle D$ are exatly the units of $\displaystyle \mathbb{R}.$ suppose that $\displaystyle -x^2+y^5=f(x)g(x),$ for some non-units $\displaystyle f,g \in D.$ then

    $\displaystyle \deg f(x)=\deg g(x)=1,$ i.e. $\displaystyle f(x)=ax+b, \ g(x)=cx+d,$ where $\displaystyle a,b,c,d \in D.$ now $\displaystyle -x^2+y^5=(ax+b)(cx+d)$ will give us $\displaystyle ac=-1, \ ad+bc=0, \ bd=y^5$ and so $\displaystyle (ad)^2=y^5,$

    which is obviously impossible because the degree of $\displaystyle (ad)^2 \in D$ is an even number.
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  3. #3
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    Thanks

    Thank you that helps a lot.
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