Lemma of Hensel

In Mathematical, the lemma of Hensel, which owes its name to the mathematician of the beginning of the twentieth century Kurt Hensel, can be seen like an analog of the Méthode of Newton in the rings of valuation supplements, i.e. it makes it possible to give approximations of the roots of the polynomials. The typical example is that of the ring of the polynomials with coefficients in $\ mathbb {Z} _p$ , the ring of the p-adic whole , for p a prime number.

Statements

Lemma of Hensel version 1

If there exists $\ alpha_0 \ in \ mathbb {Z} _p$ such as:

(I) $P (\ alpha_0) \ equiv 0 \ pmod p$ ,

(II) $P' (\ alpha_0) \ not \ equiv 0 \ pmod p$ ,

then, there exists $\ alpha \ in \ mathbb {Z} _p$ such as $P (\ alpha) = 0$ and $\ alpha \ equiv \ alpha_0 \ pmod p$ .

Lemma of Hensel version 2

If there exists $\ alpha_0 \ in \ mathbb {Z} _p$ such as, for a certain entirety $N$ , one has:

(I) $P' (\ alpha_0) \ equiv 0 \ pmod {p^N}$ ,

(II) $P' (\ alpha_0) \ not \ equiv 0 \ pmod {p^ {N+1}}$ ,

(III) $P (\ alpha_0) \ equiv 0 \ pmod {p^ {2N+1}}$ ,

then, there exists $\ alpha \ in \ mathbb {Z} _p$ such as $P (\ alpha) = 0$ and $\ alpha \ equiv \ alpha_0 \ pmod {p^ {N+1}}$ .

Category: Algebraic theory of the numbers

Random links: