examples for Hensel’s lemma

Example 1.

Let $p$ be a prime number greater than $2$. Are there solutions to $x^{2}+7=0$ in the field $\mathbb{Q}_{p}$ (the $p$-adic numbers (http://planetmath.org/PAdicIntegers))? If there are, $-7$ must be a quadratic residue modulo $p$. Thus, let $p$ be a prime such that

 $\left(\frac{-7}{p}\right)=1$

where $(\frac{\cdot}{p})$ is the Legendre symbol. Hence, there exist $\alpha_{0}\in\mathbb{Z}$ such that $\alpha_{0}^{2}\equiv-7\mod p$. We claim that $x^{2}+7=0$ has a solution in $\mathbb{Q}_{p}$ if and only if $-7$ is a quadratic residue modulo $p$. Indeed, if we let $f(x)=x^{2}+7$ (so $f^{\prime}(x)=2x$), the element $\alpha_{0}\in\mathbb{Z}_{p}$ satisfies the conditions of the (trivial case of) Hensel’s lemma. Therefore there exist a root $\alpha\in\mathbb{Q}_{p}$ of $x^{2}+7=0$.

Example 2.

Let $p=2$. Are there any solutions to $x^{2}+7=0$ in $\mathbb{Q}_{2}$? Notice that if we let $f(x)=x^{2}+7$, then $f^{\prime}(x)=2x$ and for any $\alpha_{0}\in\mathbb{Z}_{2}$, the number $f^{\prime}(\alpha_{0})=2\alpha_{0}$ is congruent to $0$ modulo $2$. Thus, we cannot use the trivial case of Hensel’s lemma.

Let $\alpha_{0}=1\in\mathbb{Z}_{2}$. Notice that $f(1)=8$ and $f^{\prime}(1)=2$. Thus

 $|8|_{2}<|2^{2}|_{2}$

and the general case of Hensel’s lemma applies. Hence, there exist a $2$-adic solution to $x^{2}+7=0$. The following is the $2$-adic canonical form (http://planetmath.org/PAdicCanonicalForm) for one of the solutions:

 $\alpha=1+1\cdot 2^{3}+1\cdot 2^{4}+\ldots=\ldots 11001$
Title examples for Hensel’s lemma ExamplesForHenselsLemma 2013-03-22 15:08:36 2013-03-22 15:08:36 alozano (2414) alozano (2414) 6 alozano (2414) Example msc 12J99 msc 11S99 msc 13H99