Question

Factor the polynomial $3 x^3+x^2+11 x+1$ over the field $G F_{17}$.

    Factor the polynomial $3 x^3+x^2+11 x+1$ over the field $G F_{17}$.

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Darel W. Hardy, Fred… 2nd Edition

Chapter 9, Problem 12

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Step 1

To do this, we substitute $x$ with each element of $GF_{17}$ (i.e., the integers from 0 to 16) and see if the polynomial evaluates to zero. Show more…

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Factor the polynomial $3 x^3+x^2+11 x+1$ over the field $G F_{17}$.

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Key Concepts

Finite Fields

A finite field is a field with a finite number of elements in which every non-zero element has a multiplicative inverse. Operations such as addition, subtraction, multiplication, and division (except by zero) are performed modulo a prime number, as in GF17, where the elements are {0, 1, ..., 16}.

Modular Arithmetic

Modular arithmetic involves performing calculations where numbers wrap around after reaching a certain value—the modulus. In the context of GF17, all operations are carried out modulo 17, ensuring that results remain within the field.

Polynomial Factorization over Finite Fields

Factoring a polynomial over a finite field involves expressing it as a product of lower degree polynomials with coefficients in that field. Techniques often include checking for roots within the field, applying the factor theorem, and using polynomial division adapted to the finite field setting.

Factor Theorem

The factor theorem is a fundamental concept that states if a polynomial f(x) has a root r in a given field, then (x - r) is a factor of f(x). This principle is applied when factoring polynomials over finite fields by testing potential roots.

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