Question

Use the Extended Euclidean Algorithm to find \(\operatorname{ged}(f, g)\), for \(f, g \in \mathbb{Z}_{p}[x]\) in each of the following examples (arithmetic in \(\mathrm{Z}_{p}=\\{0, \ldots, p-1\\}\) is done modulo \(p\) ). In each case compute the corresponding polynomials \(s\) and \(t\) such that \(\operatorname{gcd}(f, g)=s f+t \mathrm{~g}\). (i) \(f=x^{3}+x+1, g=x^{2}+x+1\) for \(p=2\) and \(p=3\). (ii) \(f=x^{4}+x^{3}+x+1, g=x^{3}+x^{2}+x+1\) for \(p=2\) and \(p=3\). (iii) \(f=x^{5}+x^{4}+x^{3}+x+1, g=x^{4}+x^{3}+x^{2}+x+1\) for \(p=5\). (iv) \(f=x^{5}+x^{4}+x^{3}-x^{2}-x+1, g=x^{3}+x^{2}+x+1\) for \(p=3\) and \(p=5\).

Short answer

Use the Extended Euclidean Algorithm to find the gcd of polynomials in \(\mathbb{Z}_p[x]\) and express it as a combination of the given polynomials.

Step-by-step solution

Understanding the Problem

We need to find the greatest common divisor (gcd) of two polynomials using the Extended Euclidean Algorithm in the polynomial ring \(\mathbb{Z}_p[x]\), where arithmetic is done modulo \(p\). Additionally, we must express the gcd as a linear combination of the given polynomials \(f\) and \(g\).

Simplifying and Computing with Modulo Arithmetic

To simplify each problem, we start by substituting the given polynomials \(f\) and \(g\) with their respective coefficients, and perform arithmetic in \(\mathbb{Z}_p[x]\). In all cases, ensure to take each coefficient modulo \(p\) after arithmetic operations.

Applying the Euclidean Algorithm

For each pair \((f, g)\) in the given examples, apply the Euclidean algorithm by performing polynomial division. At each step, compute the remainder \(r(x)\) when dividing \(f(x)\) by \(g(x)\). Then replace \(f\) with \(g\) and \(g\) with \(r\), continuing until the remainder is zero. The last nonzero remainder is the gcd.

Utilizing the Extended Euclidean Algorithm

While applying the Euclidean Algorithm, simultaneously calculate the polynomials \(s(x)\) and \(t(x)\). Record each step to keep track of how the preceding remainders can be expressed in terms of \(f\) and \(g\). This involves back-substitution similar to the Euclidean Algorithm for integers.

Example Case (i),\(p=2\)

For example (i) with \(p=2\), the polynomials are \(f=x^3+x+1\) and \(g=x^2+x+1\). The Euclidean algorithm yields \(r(x)\) such that \(r(x)=1\) (constant), leading us to gcd \(1\). The Extended Euclidean Algorithm provides polynomials \(s(x)\) and \(t(x)\) such that \(sf + tg = 1\). Solve to find \(s(x)\) and \(t(x)\) accordingly.

Verify and Cross-check Answers

Once calculations are complete for each pair, verify solutions by checking gcd computations and ensuring \(sf + tg = \text{gcd}(f, g)\). Cross-check these with direct computations where possible to confirm accuracy.

Key concepts

Polynomial Rings

In algebra, a polynomial ring is a set of polynomials with coefficients in a given ring. In the context of the exercise, we work specifically within the polynomial ring \(\mathbb{Z}_p[x]\). The symbols \(\mathbb{Z}_p\) represent a finite field containing integers modulo \(p\). This means all polynomial operations are performed with coefficients reduced by modulo \(p\).

• A polynomial ring consists of expressions formed by adding and multiplying these polynomials.
• Just like numbers, polynomials in this ring can be added, subtracted, and multiplied, but each coefficient is considered modulo \(p\).
• This system allows us to leverage the properties of modulo arithmetic in computations.

Considering polynomial operations in the ring \(\mathbb{Z}_p[x]\) is crucial since it determines how polynomial arithmetic is performed, particularly important when applying algorithms like the Euclidean Algorithm.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) of two polynomials is the largest polynomial that divides both without leaving a remainder. Finding the GCD of polynomials is analogous to finding the GCD of integers.

• The Euclidean Algorithm, a method for finding the GCD, can be extended to polynomials.
• This involves continually dividing the polynomials and taking the remainder until a remainder of zero is reached.
• The non-zero remainder just before the last division is the GCD.

This is powerful because it allows us to simplify polynomial expressions and solve polynomial equations more easily by removing common factors. The GCD is not just a measure of shared polynomial factors; it also applies in expressing polynomials through linear combinations of other polynomials, a concept utilized in the Extended Euclidean Algorithm.

Modulo Arithmetic

Modulo arithmetic is a system of arithmetic where numbers "wrap around" upon reaching a certain value, known as the modulus \(p\). In the polynomial context, it means working with the coefficients of polynomials "modulo \(p\)".

• In \(\mathbb{Z}_p\), every integer coefficient in the polynomial is considered under modulo \(p\).
• For instance, in \(\mathbb{Z}_2\), the coefficients are either 0 or 1, reducing any integer to its remainder when divided by 2.
• Modulo arithmetic ensures computations remain within the finite set of numbers, making calculations simpler and more systematic.

When integrating modulo arithmetic with polynomial calculations, especially within the Euclidean algorithm, we focus on simplifying polynomial coefficients after every operation. This ensures the calculations remain correct under the modulus and aligns with operations in fields like \(\mathbb{Z}_p[x]\).

Math Fields (Z_p)

Math Fields such as \(\mathbb{Z}_p\) are defined as a set of numbers where arithmetic operations such as addition, subtraction, multiplication, and division (except by zero) are possible.

• \(\mathbb{Z}_p\) indicates a finite field for a prime number \(p\). For example, \(\mathbb{Z}_2\) is a field consisting of elements \{0, 1\}.
• Fields ensure that every non-zero element has a multiplicative inverse, a key property used in solving equations.
• Using these properties, polynomial arithmetic in \(\mathbb{Z}_p[x]\) becomes structured and allows for operations like polynomial division to proceed smoothly.

Understanding Math Fields is important when tackling problems of polynomial GCD because they set the framework for operations. Each polynomial operation in \(\mathbb{Z}_p[x]\) respects the rules of the field, making them predictable and ensuring consistent results in algorithms like the Extended Euclidean Algorithm.