TITLE: Why is it that $x^4+2x^2+1$ is reducible in $\mathbb{Z}[x]$ but has no roots in $\mathbb{Q}$?
QUESTION [7 upvotes]: $\textbf{ Lemma:}$  A non constant primitive polynomial $f(x) \in \mathbb{Z}[x]$ is irreducible in $\mathbb{Q}[x]$ if and only if $f(x)$ is irreducible in $\mathbb{Z}[x]$.
I am reading a book in which it is given that $f(x)=x^4+2x^2+1$ is primitive in $\mathbb{Z}[x]$ and it has no roots in $\mathbb{Q}$ but it is reducible over $\mathbb{Z} $ as $f(x)=(x^2+1)(x^2+1)$.
My question is doen't it conradict this lemma?  since irreducible over $\mathbb{Z}$ and irreducible over $\mathbb{Q}$ is the same thing for primitive polynomial.

REPLY [2 votes]: It is not necessary that a polynomial is reducible over a field iff it has a root in field. By meaning of a reducible polynomial is that we express the polynomial into product of non constant polynomials over the field. And a result that a polynomial  having degree 2 or 3 is reducible iff it has root in field.
In your example  degree of polynomial is 4.