Question

Let \(q\) be a prime power, \(t \in \mathbb{N}\) a prime divisor of \(q-1\), and \(a \in \mathbb{F}_{q}^{\times}\). (i) Show that the polynomial \(x^{t}-a \in \mathbb{F}_{q}[x]\) splits into linear factors if \(a\) is a \(t\) th power (ii) Show that \(x^{t}-a\) is irreducible if \(a\) is not a \(t\) th power Hint: Use (i) for the splitting field of \(x^{t}-a\) and consider the constant coefficient of a hypothetical factor \(f \in \mathbb{F}_{q}[x]\) of \(x^{t}-a\). (iii) Derive a formula for the probability that a random binomial \(x^{t}-a\) (that is, for random \(a \in \mathbb{F}_{q}^{\times}\)) is irreducible, and compare it to the probability that a random polynomial of degree \(t\) in \(\mathbb{F}_{q}[x]\) is irreducible.

Short answer

(i) It splits into linear factors if \( a \) is a \( t \)-th power. (ii) Irreducible if \( a \) is not a \( t \)-th power. (iii) Probability \( \frac{t-1}{t} \), differs from random degree \( t \) polynomial probability \( \frac{1}{t} \).

Step-by-step solution

Understanding Prime Power and Polynomial

Given that \( q \) is a prime power, \( t \) is a prime divisor of \( q-1 \), and \( a \in \mathbb{F}_{q}^{\times} \), we need to analyze how the polynomial \( x^t - a \) behaves based on whether \( a \) is a \( t \)-th power or not.

Analyzing a as a t-th Power

If \( a \) is a \( t \)-th power, then \( a = b^t \) for some \( b \in \mathbb{F}_q^{\times} \). In this case, the polynomial \( x^t - a \) can be factored as follows: \[ x^t - a = x^t - b^t = (x-b)(x - \, \omega b) \ldots (x - \, \omega^{t-1} b) \] where \( \omega \) is a primitive \( t \)-th root of unity. This shows \( x^t - a \) splits into linear factors.

Polynomials for Non t-th Power a

When \( a \) is not a \( t \)-th power, assume \( f(x) \) is a factor of \( x^t - a \). Consider the field extension \( \mathbb{F}_q(\sqrt[t]{a}) \), making \( \alpha = \sqrt[t]{a} \) one of its roots. If \( f(x) \) is reducible, then all roots must be within \( \mathbb{F}_q \) contradicting the minimum extension degree required by non-\( t \)-power. Thus, \( x^t - a \) is irreducible.

Probability of Irreducibility

The occurrence of \( a \) as a non-\( t \)-th power happens \( \frac{q-1}{t} \) times. Hence the probability that a random \( a \in \mathbb{F}_{q}^{\times} \) is not a \( t \)-th power is \( \frac{t-1}{t} \). Comparing with the count of all degree \( t \) irreducible polynomials, which by a counting formula, occurs with probability roughly \( \frac{1}{t} \), shows that these probabilities can vary.

Key concepts

Irreducible Polynomials

Irreducible polynomials are a fundamental concept in finite fields and algebra. A polynomial is considered irreducible over a given field if it cannot be factored into the product of lower-degree polynomials with coefficients in that field.
In the context of the exercise, we encounter the polynomial \( x^t - a \). This polynomial can either be reducible or irreducible depending on whether \( a \) is a \( t \)-th power in the field \( \mathbb{F}_q \).

• If \( a \) is not a \( t \)-th power, the polynomial \( x^t - a \) does not decompose further in the given field, showing its irreducibility.
• An irreducible polynomial like \( x^t - a \) must remain whole in any attempted factorization within the same field, without splitting into simpler polynomials.

Understanding irreducibility is key to polynomial factorization, as it helps determine how complex expressions can be manipulated or simplified within mathematical and algebraic operations.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. Finding these smaller polynomials involves understanding both the coefficients and the degree of each term in the original expression.
For the exercise's polynomial \( x^t - a \), knowing whether \( a \) is a \( t \)-th power dictates the factorization:

• If \( a = b^t \) for some \( b \) in the field, the polynomial can be factored. This is because it essentially represents a power difference, \( x^t - b^t \), and can be expressed as linear factors \((x-b), (x - \, \omega b), ...., (x - \, \omega^{t-1} b)\).
• \( \omega \) here is a primitive \( t \)-th root of unity, symbolizing the rotational symmetry that arises in these expansions.

By factorizing such polynomials, mathematicians gain insight into the structure of the roots and the field extensions necessary for solving equations involving these polynomials. It essentially transforms a complex problem into simpler, more manageable parts.

Prime Power

A prime power is a term used to describe numbers of the form \( q = p^n \), where \( p \) is a prime number and \( n \) is a positive integer. Prime powers are integral parts of understanding fields, particularly finite fields or \( \mathbb{F}_q \).
In the exercise, \( q \) as a prime power is essential because it determines the structure and properties of the field \( \mathbb{F}_q \). The properties of \( \mathbb{F}_q \) depend crucially on \( q \) as:

• The number of elements in a field \( \mathbb{F}_q \) is exactly \( q \), which is crucial for processes like identifying roots of polynomials and determining irreducibility.
• The relationship of \( t \) being a divisor of \( q-1 \) ties into cyclic group properties, important for establishing when a polynomial like \( x^t - a \) can be split or remains irreducible.\

Understanding prime powers provides a foundation for further exploring finite fields and studying polynomials within these algebraic structures.