Timeline for Square root in number field

6 events

when toggle format	what		by	license	comment
Oct 22, 2023 at 9:30	vote	accept	Tippisum
Jul 8, 2022 at 9:15	comment	added	François Brunault		Q1: Finding roots of (or factoring) polynomials over number fields is easy computationally, there are polynomial time algorithms, e.g. in PARI/GP there is Belabas's variant of van Hoeij's algorithm. So you can just factor or find the roots of $x^2-b$ over $\mathbb{Q}(\alpha)$.
Jul 1, 2022 at 12:21	comment	added	Emil Jeřábek		(1) If you can compute square roots of numbers that are known to be squares, you can also determine whether a given element $b$ is a square or not: just run the square root algorithm on $b$, and check if the result $x$ (if any) satisfies $x^2=b$. If $b$ is a square, then the algorithm is guaranteed to compute a square root of $b$, thus the test is positive. If $b$ is not a square, then either the algorithm fails due to running into some sort of internal inconsistency, or it computes a garbage number $x$ whose square in any case cannot be $b$, as $b$ is not a square. Thus, the test is negative.
Jun 30, 2022 at 18:19	answer	added	R.P.		timeline score: 7
Jun 29, 2022 at 17:22	comment	added	Gro-Tsen		Concerning question (1), this may not be the most efficient thing to do, but you can certainly apply algorithms for factoring polynomials over number fields (Googling “factoring polynomials over number fields” returns many relevant results) to factor $X^2 - b$ in $\mathbb{Q}(\alpha)[X]$.
Jun 29, 2022 at 15:55	history	asked	Tippisum	CC BY-SA 4.0