I need a criteria how to determine if an element x is a square in ℤ₂ (2-adic numbers).
Asked
Active
Viewed 79 times
2
-
1Welcome to MSE. Please use MathJax. – José Carlos Santos Sep 20 '17 at 10:58
-
1$\mathbb Z_2$ is misleading. Usually, this means the group with $2$ elements. Additionally, is $x$ an integer or only rational ? – Peter Sep 20 '17 at 11:14
-
3@Peter Usually $\Bbb Z_2$ denotes the 2-adic integers. The cyclic group with 2 elements should possibly be written $Z_2$, $\Bbb Z/2\Bbb Z$, or $\Bbb Z/(2)$. The fact that it is often also written $\Bbb Z_2$ is rather confusing (at least to me) cause there is no common alternative to using $\Bbb Z_2$ for 2-adics ... – Hagen von Eitzen Sep 20 '17 at 11:23
-
Do you know Hensel's lemma? – Hagen von Eitzen Sep 20 '17 at 11:25
-
1@Peter $2$-adic numbers are not real numbers. See https://en.wikipedia.org/wiki/P-adic_number – Crostul Sep 20 '17 at 11:48
-
Possible duplicate of Characterization of integers which has a $2$-adic square root – Crostul Sep 20 '17 at 11:50
-
@Crostul Thank you for clarifying that. – Peter Sep 20 '17 at 12:16
-
Joanna, welcome to Math.SE. Unfortunately your question was put on hold by popular vote. By studying this page from the Help Center you can figure out what is bothering the users who voted that way, and what steps you can take to make them change their mind. I have two suggestions. A) This may have been handled already on our site, did you search? B) Dealing with power of two factors is easy here, but for 2-adic units you need a modulo eight argument. – Jyrki Lahtonen Sep 20 '17 at 19:08
-
Re my suggestion A: The question Crostul linked to is very useful to you. Nominally it is about rational integers only, but.... – Jyrki Lahtonen Sep 20 '17 at 19:11
-
thank you all very much. I have a lot to learn here – joanna Sep 21 '17 at 06:39