whether a polynomial is irreducible or not

Question

Let $$I=\langle x^4+3x^2+2\rangle$$ and $F=\mathbb{Q}[x]/I$

The question is whether $F$ is a field or not.

I know how to proceed this problem. I'm confusing with irreducibility of the given polynomial

Here, $x^4+3x^2+2=0$ implies $x^4+3x^2=-2$. This happens only $x \in \mathbb{C}$. So the polynomial has no root in $\mathbb{Q}$ and hence irreducible.

However, $x^4+3x^2+2=(x^2+2)(x^2+1)$, so the given polynomial is reducible.

Where I'm doing wrong ?

Answer 1

Irreducibility over $\Bbb Q$ does not mean "has no rational root". As this example shows, a polynomial may be written as a non-trivial product but with no roots over $\Bbb Q$. The smallest degree where his happens is $4$ where a polynomial, as here, can be a product of two irreducible quadratics.

Answer 2

You are assuming that $p(x)$ has no roots in $\mathbb Q$ implies that $p(x)$ is irreducible in $\mathbb Q$. This is true if the degree of $p(x)$ is $2$ or $3$, but not in general, as your own example shows.