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(For example, in the case of primitive roots, there is basic field theory combined with Hensel's lemma/Newton approximation; this style of argument extends, in some form, to the very general setting of …

To show it is sufficient, a lemma of Davenport and Cassels, using Hasse-Minkowski, shows that $a$ is the sum of three rational squares. … Complete the analogy: Hensel's lemma is to Newton's method as this technique is to _____________________. …

Lemma and the primitive element theorem. … By using Hensel's Lemma in finite local $R$-algebras, to give a map from a finite \'etale $R$-algebra $A$ to a finite $R$-algebra $B$ is the same as to give a map $A_0 \rightarrow B_0$ between their special …

If its eigenvalues mod $x$ are all distinct, we are done, because we can find roots of its characteristic polynomial in $\mathbb R[[x]]$ by Hensel's lemma. … Lemma: Let $M$ be a symmetric matrix over $\mathbb R[[x]]$ such that some eigenvalues are distinct mod $x$. …

.$ Using the Chinese Remainder Theorem and Hensel's Lemma one can quickly reduce this claim to the case that $n=p$ is a prime. …

The idea is that this field is not complete but is Henselian -- it satisfies the conclusion of Hensel's Lemma. …

By Hensel's Lemma, the square of a prime $p\geq 5$ divides $\Phi_{24}(n)$ for some $n$ if and only if $p$ divides $\Phi_{24}(m)$ for some $m$. …

Appearances can sometimes be deceiving: automorphic numbers may at first look totally recreational, but they are connected with the Chinese remainder theorem, Hensel's lemma, and the contraction mapping …

but not too difficult (Nakayama's lemma, Hensel's lemma, Sperner's lemma). … However, there do exist "high-powered" examples such as the fundamental lemma or the Szemerédi regularity lemma. …

He says that this result is today called Hensel's Lemma and that Hensel's standard proof applies. … It was my initial thought that the Hensel-Kurschak Lemma would follow easily from one of the more standard forms of Hensel's Lemma. …

example, the Diophantine equation $(x^2 - 2)(x^2 - 3)(x^2 - 6) = 0$ has this property: for any prime $p$, at least one of $2, 3, 6$ must be a quadratic residue, so there is a solution $\bmod p$, and by Hensel's … lemma (which has to be applied slightly differently when $p = 2$) there is a solution $\bmod p^n$ for any $n$. …

lemma"); this is what makes localization work for sheaves on the small Nisnevich or étale sites. … The key input here is a geometric presentation lemma of Gabber, a statement of which can be found as Lemma 15 in the introduction to Morel's book (http://www.mathematik.uni-muenchen.de/~morel/Prepublications …

Hensel's lemma comes with hypotheses. …

In my class today, I presented the "Omnibus Hensel's Lemma". Part a) was: the following five conditions on a valued field are all equivalent. … For instance, the origin of the question I cited above was the fact that in Tuesday's class I stpdly chose the wrong form of Hensel's Lemma to use to try to deduce yet another version of Hensel's Lemma …

Of course having an isomorphism in this case depends on Zorn's lemma, so field isomorphisms $\mathbf C_p \to \mathbf C$ are totally nonconstructive. … Thanks to Hensel's lemma and finiteness of the residue field of $(K,|\cdot|)$, we can describe the group $\mathcal O_K^\times$ purely algebraically: $$ \mathcal O_K^\times = \{x \in K^\times : x {\sf \ …