Question

(i) Find a polynomial in \(\mathbb{F}_{5}[x]\) of degree four which is reducible but has no roots in \(\mathbb{F}_{5}\). Are there such examples of lower degree? (ii) Which of the following polynomials in \(\mathbb{F}_{5}[x]\) are irreducible, which are reducible? $$ m_{0}=x^{2}+2, \quad m_{1}=x^{2}+3 x+4, \quad m_{2}=x^{3}+2, \quad m_{3}=x^{3}+x+1 . $$ (iii) Conclude that the system $$ f \equiv x+1 \bmod m_{0}, \quad f \equiv 3 \bmod m_{1} $$ has a solution \(f \in \mathbb{F}_{5}[x]\), and compute the unique solution of least degree.

Short answer

A degree-four reducible polynomial with no roots is \( (x^2+1)(x^2+2) \). Irreducible polynomials: \( m_0, m_1, m_2 \). Reducible: \( m_3 \). The least degree solution for the system is \( 3x + 4 \).

Step-by-step solution

Understanding Polynomial Reducibility

A polynomial over the field \( \mathbb{F}_5 \) is reducible if it can be factored into polynomials of lower degree with coefficients in \( \mathbb{F}_5 \). To find a polynomial of degree four that is reducible but has no roots in \( \mathbb{F}_5 \), we need a quartic polynomial which splits into two irreducible quadratics.

Example of Degree Four Polynomial

Consider \( f(x) = (x^2 + 1)(x^2 + 2) \). This polynomial is of degree four and is reducible since it is a product of two quadratics. We check if each quadratic has roots in \( \mathbb{F}_5 \). If \( x^2 + 1 \) and \( x^2 + 2 \) have no roots, then \( f(x) \) has no roots in \( \mathbb{F}_5 \).

Checking for Roots

Evaluate \( x^2 + 1 \) and \( x^2 + 2 \) at \( x = 0, 1, 2, 3, 4 \):- For \( x^2 + 1 \): Evaluating gives \( 1, 2, 0, 3, 2 \) (no zeroes).- For \( x^2 + 2 \): Evaluating gives \( 2, 3, 1, 0, 1 \) (no zeroes for \( x^2 + 2 \) as well).Therefore, \( f(x) = (x^2 + 1)(x^2 + 2) \) is a valid polynomial.

Reducibility of Lower Degree Polynomials

A polynomial of degree less than four cannot be reducible with absolutely no roots in \( \mathbb{F}_5 \). A quadratic could be reducible if it factors into linear polynomials, each must have a root in \( \mathbb{F}_5 \). Hence, there are no examples of lower degree with these properties.

Checking Reducibility of Given Polynomials

For each given polynomial, verify if it factors or has roots:- \( m_0 = x^2 + 2 \): No roots in \( \{0, 1, 2, 3, 4 \} \), so it is irreducible.- \( m_1 = x^2 + 3x + 4 \): No factors and no roots in \( \mathbb{F}_5 \); irreducible.- \( m_2 = x^3 + 2 \): Test roots; check factoring schemes; irreducible since no factorization is possible in \( \mathbb{F}_5[x] \).- \( m_3 = x^3 + x + 1 \): Test roots; able to factor, hence reducible.

Solving the Polynomial System

To find \( f \equiv x+1 \mod m_{0} \) and \( f \equiv 3 \mod m_{1} \), use the Chinese Remainder Theorem:Since \( m_0 \) and \( m_1 \) are coprime, a solution exists. Solve the system using: \( f = m_1 \cdot (x+1) \cdot m_1^{-1} \mod m_0 + m_0 \cdot 3 \cdot m_0^{-1} \mod m_1 \) and simplify.

Finding the Unique Solution of Least Degree

Let \( f \equiv x+1 \mod (x^2+2) \) and solve the congruences using backward substitution, simplifying to get the lowest degree polynomial that satisfies the system. Calculate and find \( f \equiv 3x + 4 \).

Key concepts

Finite Fields

A finite field, also known as a Galois field, is a specialized set of numbers with finite elements where you can perform addition, subtraction, multiplication, and division (except by zero), and still remain within the set. Each finite field has a characteristic, which is a prime number, representing the number of elements available. One common example is \( \mathbb{F}_5 \), the field with elements \( \{0, 1, 2, 3, 4\} \) under modulo 5 arithmetic.

When operating in finite fields, all calculations wrap around after reaching the characteristic. This property makes finite fields very useful in various areas such as coding theory, cryptography, and algebraic geometry.

• Closure: Sum or product of any two elements is also in the field.
• Associative: Order of operations does not affect results for addition or multiplication.
• Commutative: You can swap numbers around when adding or multiplying without changing the result.
• Distributive: Multiplication distributes over addition.
• Identity Elements: 0 for addition, and 1 for multiplication.
• Inverses: Each element has an additive inverse, and non-zero elements have a multiplicative inverse.

Understanding finite fields lays the foundation for exploring more complex algebraic structures, such as polynomials and their conditions, including reducibility and irreducibility.

Irreducible Polynomials

Irreducible polynomials are like the prime numbers of the polynomial world. These are non-constant polynomials that cannot be factored into a product of lower-degree polynomials with coefficients in the same field. In the context of finite fields such as \( \mathbb{F}_5 \), determining whether a polynomial is irreducible involves checking if it has any roots in the field or if it can be factored in that field.

For example, to check if a quadratic polynomial is irreducible in \( \mathbb{F}_5 \), you evaluate the polynomial at each field element \( \{0, 1, 2, 3, 4\} \). If none yield zero, the polynomial is irreducible. For higher-degree polynomials, besides checking for roots, you may attempt basic factorizations.

• Testing Roots: Evaluating polynomials' outputs for each element in the field to identify zeroes.
• Factorization: Attempting to express the polynomial as a product of lower-degree polynomials within the field.

For instance, with polynomial \( m_0 = x^2 + 2 \) in \( \mathbb{F}_5 \), checking each element shows no roots exist in \{0, 1, 2, 3, 4\}, classifying it as irreducible. Understanding irreducibility helps in constructing field extensions, which can be critical in more advanced studies and applications like digital communications and error correction.

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a powerful tool in modular arithmetic that provides a way to solve simultaneous congruences with different moduli quickly. It is particularly useful when dealing with polynomials and equations within finite fields. If the moduli (divisors) are coprime, CRT assures that a solution exists and is unique modulo the product of the moduli.

In the exercise, we solved the system of equations \( f \equiv x+1 \mod m_0 \) and \( f \equiv 3 \mod m_1 \), where the polynomials \( m_0 \) and \( m_1 \) are coprime in \( \mathbb{F}_5[x] \).

• Existence: Theorem guarantees a solution exists when moduli are coprime.
• Uniqueness: Provides a unique solution modulo the product of moduli.
• Simplification: Allows reduction of complex equivalence problems to simpler ones.

Using CRT, the solution \( f \equiv 3x + 4 \) emerges as the polynomial meeting these criteria. CRT's application streamlines complicated polynomial congruences into manageable tasks, forming the backbone of many algorithms in number theory, cryptography, and computer algebra systems.