Thread: 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.

Originally Posted by Algebraicgeometry421

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 as an element of where note that the units of are exatly the units of suppose that for some non-units then

i.e. where now will give us and so

which is obviously impossible because the degree of is an even number.

Thank you that helps a lot.