Thread: Irreducible polynomial

Hi everybody,

Could You help me with the following statement:

Proof that there are (one variable) polynomials of any natural degree with rational coefficients, which are irreducible over the field of rational numbers.

Thanks.

It's a direct consequence from Eisenstein's criterion.

Originally Posted by Migotek84

Hi everybody,

Could You help me with the following statement:

Proof that there are (one variable) polynomials of any natural degree with rational coefficients, which are irreducible over the field of rational numbers.

Thanks.

In other words, show that, given any n> 0, there exist a polynomial of degree n which has no rational root.

I would suggest looking at $\displaystyle x^n- p= 0$ where p is a prime number.

Originally Posted by HallsofIvy

In other words, show that, given any n> 0, there exist a polynomial of degree n which has no rational root.

"no root" implies "irreducible" is true only if n<=3. It is a really common mistake. However your example still works due to Eisenstein's criterion.