Euler's remarkable '-producing polynomial and quadratic UFDs Example of

THEOREM.
The polynomial ${\displaystyle f\left(x\right)=(x-\alpha )(x-{\alpha }^{\prime })=x^2+x+k}$ assumes only ' values for ${\displaystyle 0\le x\le k-2\iff Z\left[\alpha \right]}$ is a PID.
HINT.
Show all 's $rm p\le \sqrt{n},;n=1-4k$ satisfy $rm \left(\frac{n}{p}\right)=-1$ so no 's split/ramify.
For proofs, see e.g. Cohn, Advanced Number Theory, pp. 155-156, or Ribenboim, My numbers, my friends, 5.7 p.108. Note: both proofs employ the bound $rm p<\sqrt{n}$ without explicitly mentiioning that this is a consequence of the Minkowski bound - presumably assuming that is obvious to the reader based upon earlier results. Thus you'll need to read the prior sections on the Minkowski bound. Compare Stewart and Tall, Algebraic number theory and FLT, 3ed, Theorem 10.4 p.176 where the use of the Minkowski bound is mentioned explicitly.
Alternatively see the self-contained paper [1] which proceeds a bit more simply, employing Dirichlet approximation to obtain a generalization of the Euclidean algorithm (the Dedekind-Rabinowitsch-Hasse criterion for a PID). If memory serves correct, this is close to the approach originally employed by Rabinowitsch when he first published this theorem in 1913.