Question

You are to show that for a prime power \(q\) and a positive integer \(n \geq 2\), the probability for a random polynomial in \(\mathbb{F}_{q}[x]\) of degree \(n\) to be squarefree is \(1-1 / q\). Let \(s_{n}\) denote the number of monic squarefree polynomials of degree \(n\) in \(\mathbb{F}_{q}[x]\). Then \(s_{0}=1\) and \(s_{1}=q\). (i) Prove the recursive formula \(\Sigma_{0 \leq 2 k \leq n} q^{k} s_{n-2 k}=q^{n}\). Hint: Every monic polynomial \(f \in \mathbb{F}_{q}[x]\) can be uniquely written as \(f=g^{2} h\) with a squarefree monic polynomial \(h\). (ii) Conclude that \(s_{n}=q^{n}-q^{n-1}\) if \(n \geq 2\) by subtracting a suitable multiple of the above formula for \(n-2\) from the formula itself.

Short answer

The probability is \(1 - \frac{1}{q}\). Squarefree polynomial count is \(s_n = q^n - q^{n-1}\).

Step-by-step solution

Understand the Problem Statement

We need to show that a random monic polynomial of degree \(n\) in \(\mathbb{F}_{q}[x]\) is squarefree with probability \(1 - \frac{1}{q}\). A polynomial is squarefree if it has no repeated factors.

Define Monic Polynomials and Their Count

Monic polynomials are those where the leading coefficient is 1. For a given degree \(n\), there are \(q^n\) monic polynomials in \(\mathbb{F}_{q}[x]\). This is because each of the \(n\) coefficients, except the leading one, can independently be chosen from \(q\) possibilities.

Recursive Formula

We use the hint that any monic polynomial \(f\) in \(\mathbb{F}_{q}[x]\) can be expressed as \(f = g^2 h\), where \(h\) is a squarefree monic polynomial. We calculate the summation \(\Sigma_{0 \leq 2k \leq n} q^{k} s_{n - 2k} = q^n\). This equation accounts for all monic polynomials by appropriately combining squares \(g^2\) and squarefree parts \(h\).

Verification of the Formula

For each \(k\), there are \(q^k\) choices for \(g\), a monic polynomial of degree \(k\), and \(s_{n - 2k}\) choices for \(h\), a squarefree monic polynomial of degree \(n - 2k\). The formula sums over all possible values of \(g\) and \(h\) such that their degrees fit the requirement \(n = 2k + \deg(h)\).

Simplify to Obtain \(s_n = q^n - q^{n-1}\)

We subtract the equation for \(n-2\) from the equation for \(n\) to isolate \(s_n\):\[\Sigma_{0 \leq 2k \leq n} q^k s_{n-2k} = q^n\]and\[\Sigma_{0 \leq 2k \leq n-2} q^k s_{n-2-2k} = q^{n-2}\]Leading to: \(q^{n-1}(s_{n-2} = q^{n-1})\), therefore, validating \(s_n = q^n - q^{n-1}\) for \(n \geq 2\).

Conclude by Probability

Since \(s_n\) is the count of squarefree polynomials and \(q^n\) is the total count of monic polynomials, the probability \(P\) that a random monic polynomial is squarefree is given by: \[P = \frac{s_n}{q^n} = \frac{q^n - q^{n-1}}{q^n} = 1 - \frac{1}{q}\].

Key concepts

Finite Fields

A finite field, denoted by \(\mathbb{F}_q\), is a field with a finite number of elements, typically a prime power \(q=p^m\) where \(p\) is a prime number and \(m\) is a positive integer. Finite fields are crucial in various areas of algebra and number theory, particularly in coding theory and cryptography.
The elements of a finite field \(\mathbb{F}_q\) can be thought of as numbers from 0 to \(q-1\) with arithmetic operations of addition and multiplication defined under modulo \(q\). The concept of modular arithmetic ensures that every non-zero element has a multiplicative inverse.
Finite fields are used to construct polynomial algebra, where polynomials have coefficients from \(\mathbb{F}_q\). These polynomials themselves can have various properties, such as being squarefree or monic, and are subject to operations like addition and multiplication of their coefficients within the field.

Monic Polynomials

Monic polynomials play a significant role in algebra over finite fields. A monic polynomial is a polynomial whose leading coefficient (the coefficient of the highest degree term) is equal to one. This property simplifies many algebraic manipulations and proofs.
In the context of \(\mathbb{F}_q[x]\), the set of all polynomials over the finite field \(\mathbb{F}_q\), every other coefficient in a monic polynomial can be any element from the field, providing \(q\) choices for each of those terms. If the polynomial is of degree \(n\), there are \(q^n\) possible monic polynomials, reflecting the combinatorial aspect of choosing each coefficient.
Monic polynomials are essential for counting arguments in algebraic contexts, as shown in exercises involving squarefree polynomials. Here, knowing the number of monic polynomials allows for determining the proportion that meet certain conditions, such as not having repeated factors.

Probability in Algebra

Probability in algebra, especially in the field of finite fields, often concerns selecting random polynomials and determining properties like being squarefree. A polynomial is squarefree if it has no repeated polynomial factors, akin to numbers having no repeated prime factors.
In the specific exercise, the probability of a random polynomial in \(\mathbb{F}_{q}[x]\) being squarefree helps understand the structure and distribution of such polynomials. This probability is calculated as the ratio of squarefree polynomials to all monic polynomials of a degree \(n\).
The formula derived in the exercise, \(P = 1 - \frac{1}{q}\), emphasizes the frequent occurrence of squarefree polynomials. It's fascinating as it shows that the likelihood of encountering a squarefree polynomial approaches certainty as \(q\) becomes larger. This key result blends the combinatorial counting of polynomials with fundamental probability concepts, providing a deep insight into algebraic structures.