Roots of $X^2-z$ in $2$-adics

Asked May 18 '15 at 08:35

Active May 18 '15 at 08:35

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Using Hensel's lemma is it true that $X^2-z$, for $z\in \mathbf Z$ any integer have no roots in 2-adics ?

Hensel's lemma only shows, if a root of a polynomial can be lifted to a root in $\mathbf Z_p$, so it doesn't say anything about the nonsolvability but if I look at the conditions:

If there is an $a$ s.t. $f(a)\equiv 0\mod p,\quad f'(a)\not\equiv 0\mod p$

If $p=2$, then the derivative is always zero for a polynomial of the form $X^2-z$, Is there still a root ?

asked May 18 '15 at 08:35

inequal

1,367

$X^2-1$ has a root, hasn't it? – MooS May 18 '15 at 08:39
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The list of "Related" questions in the right margin has a lot of helpful material. This question in particular is useful to you. – Jyrki Lahtonen May 22 '15 at 06:51

Roots of $X^2-z$ in $2$-adics

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