TITLE: When polynomial f(x^2) can be factored as g(x)·g(-x) ?
QUESTION [10 upvotes]: In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where $g(x)$ is also a polynomial with rational coefficients? 
Is there a computationally efficient way to identify if given polynomial $f(x)$ is such without factoring $f(x^2)$ ?

REPLY [2 votes]: (Caveat:  nothing in this response has been checked or even really thought through.)
If it's true with integral coefficients, it's true mod p for all p.  Now the question of whether f(x^2) splits as g(x)g(-x) for f in F_p[x] is the question of whether f is a norm in the quadratic extension obtained by adjoining a square root of x, which should just be a question about whether each irreducible factor p(x) appearing an odd number of times in f splits in that quadratic extension.  By quadratic reciprocity (I think) this comes down to whether each of these irreducible factors p(x) has p(0) a quadratic residue.  This is easy enough to check for lots of p.
Of course, to have any hope, you need f(0) to be a square (as an integer) so I think in practice what I'd do would be to take a long list of primes p, reduce f mod p for each p, and if f factors into p_1(x) ... p_k(x) mod p, check that each p_i(0) is a residue.  And if this keeps on happening you should gain confidence that your f(x^2) actually factors in this way.