Solving Polynomial Congruences

    We present three examples showing techniques how to solve a polynomial congruence if the modulus can be factored. The strategy is to use unique factorization, Hensel's Lemma and the Chinese remainder theorem.

Example (Solving a Polynomial Congruence I) Given Solve,

    Solution. The unique factorization of is and the derivative is  



We tabulate,






We now must solve,



The Chinese remainder theorem yields



where and are solutions to  







Therefore,

.

Example (Solving a Polynomial Congruence II) Given Solve,  

    Solution. The unique factorization of is and the derivative is



We tabulate,







We now must solve,





The Chinese remainder theorem yields the four solutions,









where and are solutions to  







Therefore,









are the four solutions to

Example (Solving a Polynomial Congruence III) Given,



Solve,  

    Solution. The unique factorization of is   and the derivative is



We tabulate,



At this point for , we see that 0, 1, 3, 6  









At this point for , we see that










At this point for , we see that 3, 10, 47, 331

In summary:






  
To finish we will use the Chinese remainder theorem   

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Cite this as:
Solving Polynomial Congruences
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/solving-polynomial-congruences.html