Question

Let $\alpha$ be a root of $x^2+1$ in $G F_9$. Construct the minimal polynomial of $\alpha+1$.

    Let $\alpha$ be a root of $x^2+1$ in $G F_9$. Construct the minimal polynomial of $\alpha+1$.

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Darel W. Hardy, Fred… 2nd Edition

Chapter 10, Problem 5

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Step 1

$GF_9$ is the finite field with $9$ elements. Since $9 = 3^2$, $GF_9$ can be constructed as $\mathbb{F}_3[x]/(p(x))$ where $p(x)$ is an irreducible polynomial of degree $2$ over $\mathbb{F}_3$. For simplicity, let's use $p(x) = x^2 + 1$, which is irreducible over Show more…

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Let $\alpha$ be a root of $x^2+1$ in $G F_9$. Construct the minimal polynomial of $\alpha+1$.

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Key Concepts

Minimal Polynomial

The minimal polynomial of an algebraic element over a given field is the unique monic irreducible polynomial with coefficients in that field for which the element is a root. It encapsulates the lowest degree algebraic relation that the element satisfies. Minimal polynomials are crucial for understanding the structure of field extensions and for performing computations within them.

Finite Fields

Finite fields, also known as Galois fields, are algebraic structures with a finite number of elements where addition, subtraction, multiplication, and division (except by zero) are defined and satisfy field axioms. They are commonly denoted as GF(q), where q is a prime power, and are fundamental in areas such as coding theory, cryptography, and algebraic geometry.

Field Extensions

A field extension is a larger field built over a given base field such that the operations of the base field are preserved. This concept allows for the inclusion of solutions to polynomial equations that do not have solutions within the base field. In the context of finite fields, constructing an extension, such as going from GF(p) to GF(p^n), is typically done by adjoining a root of an irreducible polynomial.

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