My question concerns the following result:
Theorem. Let $K$ be an algebraic number field. If $f(x) \in K[x]$ is irreducible over $K$ and has a root in every non-Archimedean completion of $K$, then $f$ is linear.
This appears to be a well-known result - a proof for $K = \mathbb{Q}$ can be found in answer to a question of mine here: The Hasse principle for bivariate cubic forms The proof crucially relies on the Chebotarev density theorem.
I have been trying to find a textbook reference for this proof, but have been unable to find one. Its corollary that an element in $K^{\times}$ is a square in $K$ if and only if it is a square in every non-Archimedean completion of $K$, is equivalent to the Hasse-Minkowski theorem for binary quadratic forms over all algebraic number fields (similarly for cubes in $K$ and the binary cubic analogy of the Hasse-Minkowski theorem). The trouble is that all the proofs of the Hasse-Minkowski theorem over all algebraic number fields that I have been able to find do not prove this result - Hasse's original paper makes reference to a related result in a paper by Hilbert (which I find unreadable), T.Y. Lam's book simply assumes the result without reference, and O'Meara proves the theorem by different means.
So my question to the community is: Is anyone aware of the existence of a textbook or published paper where the above result is proved?
Many thanks.
I answered the OP's question at the MO link he includes, but for completeness (in case someone else comes across this page) I'll give an answer here too: the theorem appears as a corollary on p. 170 of the second edition (not the first edition) of Lang's Algebraic Number Theory.
It also appears as Exercise 6.2 at the end of Cassels and Froehlich's Algebraic Number Theory and Exercise 5 in Chapter IV of the second edition of Janusz's Algebraic Number Fields (with a small error: Janusz's exercise is that for a monic irreducible polynomial in $\mathbf Z[x]$ of "arbitrary degree" there are infinitely many primes $p$ such that the polynomial has no root in $\mathbf Z/(p)$, so he clearly forgot to include the assumption that the polynomial has degree greater than $1$). The exercise in Janusz also appears as Theorem $1$ on the first page of Serre's paper here, where Serre correctly includes the condition that the degree of the polynomial is at least $2$.