4

The following is a use of eisenstein criterion that i have taken out from my lecture note.

$f(x, y) = x^4 +x^3y^2 +x^2y^3 +y$ is irreducible in Q[x, y]. This can be proved by treating Q[x,y] as (Q[y])[x] and applying the Eisenstein criterion with p = y.

However, I can't understand why i can apply eisenstein criterion this way. Why is y prime in Q[y]. How can I prove this? Thanks

user10024395
  • 1
  • One knows this in high-school, long before one learns the structural version in Andreas's answer. Namely $\ y\mid f(y)g(y) \Rightarrow, y\mid f(y),$ or $,y\mid g(y),$ is equivalent to $,f(0)g(0) = 0,\Rightarrow, f(0)=0,$ of $,g(0)=0,,$ by the Factor / Remainder Theorem. But the latter is true because the coefficient ring $\Bbb Q$ is a domain. – Bill Dubuque Apr 07 '15 at 15:01

2 Answers2

4

$\mathbf{Q}[y]/(y) \cong \mathbf{Q}$ is a field, so that $y$ is prime.

Andreas Caranti
  • 68,873
0

The ideal generated by $y$ in the PID, $\mathbf{Q}[y]$ is maximal (hence prime) as the quotient ring is the filed of rational numbers.

P Vanchinathan
  • 19,504