# 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:

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

Title | examples for Hensel’s lemma |
---|---|

Canonical name | ExamplesForHenselsLemma |

Date of creation | 2013-03-22 15:08:36 |

Last modified on | 2013-03-22 15:08:36 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 12J99 |

Classification | msc 11S99 |

Classification | msc 13H99 |