Solving Congruences and the Relationship between (p^{l-1}) and (p^{l})

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SUMMARY

The discussion centers on solving congruences of the form f(x) = 0 mod (p^l) and its relationship to f(x) = 0 mod (p^{l-1}). The participants explore the implications of having a known root modulo p and how that may influence the existence of roots modulo higher powers of p. The concept of Hensel's Lemma is referenced as a potential tool for lifting solutions from mod p to mod p^l, although the feasibility of this approach remains uncertain without additional information about the function f(x).

PREREQUISITES
  • Understanding of modular arithmetic
  • Familiarity with polynomial functions
  • Knowledge of Hensel's Lemma
  • Basic concepts of number theory
NEXT STEPS
  • Study Hensel's Lemma in detail to understand its application in lifting solutions
  • Explore the properties of polynomial roots in modular arithmetic
  • Investigate the implications of unique roots modulo p on higher powers
  • Learn about the structure of the multiplicative group of integers modulo p^l
USEFUL FOR

Mathematicians, number theorists, and students studying modular arithmetic and polynomial equations will benefit from this discussion.

mhill
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if we knew how to solve [tex]f(x)= 0 mod (p^{l-1})[/tex] (1)

could we solve then [tex]f(x)= 0 mod (p^{l})[/tex] for integer 'l'

the idea is, if it were easy to solve [tex]f(x)= 0 mod (p)[/tex] then we could easily find a solution to (1) but i do not know how to do it.
 
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It's not even clear that
[tex]f(x)\equiv0\pmod{p^l}[/tex]
has solutions, let alone that we can easily find them.
 
http://en.wikipedia.org/wiki/Hensel_Lifting

But don't just immediately click on that link! Think about the problem first. Suppose you knew that f(x) had exactly one root modulo p. (Let's say a is that root) Then isn't there a very narrow range of possibilities for a root of f(x) modulo p^2?
 

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