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 ${\alpha }_0^2\equiv -7modp$. 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

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:

$$\alpha =1+1\cdot 2^3+1\cdot 2^4+\mathrm{\dots }=\mathrm{\dots }11001$$