Problem 13 - A Computational Introduction to Number Theory and Algebra

Question

Show that for all primes \(p,\) the polynomial \(X^{4}+1\) is reducible in \(\mathbb{Z}_{p}[X]\). (Contrast this to the fact that this polynomial is irreducible in \(\mathbb{Q}[X]\), as discussed in Exercise \(16.49 .)\)

Step-by-Step Solution

Answer

Answer: Yes, the polynomial \(X^4 + 1\) is reducible for all prime numbers \(p\) in the ring \(\mathbb{Z}_p[X]\).

Step 1: Write the polynomial and analyze the problem

We need to show that the polynomial \(X^4 + 1 \in \mathbb{Z}_p[X]\) is reducible for all primes \(p\).

Step 2: Use a property of the complex numbers

We know that for the complex numbers, \(i = \sqrt{-1}\). Therefore, we can write: \(i^4 + 1 = (-1)^2 + 1 = 0\).

Step 3: Find the zeros of the polynomial in a field with characteristic p

According to Fermat's Little Theorem, for any prime number \(p\), and any integer \(a\) such that \(\text{gcd}(a,p)=1\), we have \(a^{p-1} \equiv 1 \pmod{p}\). To find the zeros of the \(X^4 + 1 \equiv 0 \pmod{p}\), we set \(X^4 \equiv -1 \pmod{p}\), which means we are looking for an element \(a \in \mathbb{Z}_p\) such that \(a^4 \equiv -1 \pmod{p}\). Taking both sides to power \(p-1\), we have \((a^4)^{p-1} \equiv (-1)^{p-1} \pmod{p}\).

Step 4: Simplify the equation and determine the conditions for reducibility

Now we can simplify the equation using Fermat's Little Theorem, \(a^{4(p-1)}\equiv 1 \pmod{p} \quad \Rightarrow \quad a^{4(p-1)} \equiv (-1)^{p-1} \equiv -1 \pmod{p}\). Since all non-negative powers of \(a\) are smaller than \(p\), this means \(4(p-1) > p-1\). We have the following cases: - If \(p = 2\), then the given polynomial \(X^4 + 1 \equiv (X^2+1)^2 \pmod{2}\), which is reducible. - If \(p > 2\) and \(p \equiv 1 \pmod{4}\), then there exists an element \(g \in \mathbb{Z}_p\) with order \(4(p-1)\), and \(g^{p-1} \equiv -1 \pmod{p}\). So, we can write \(X^4 + 1 \equiv (X - g^{(p-1)/2})(X + g^{(p-1)/2})(X + g^{(3p-3)/2})(X - g^{(3p-3)/2}) \pmod{p}\), which is reducible. - If \(p > 2\) and \(p \equiv 3 \pmod{4}\), \((-1)^{\frac{p-1}{2}} \equiv -1 \pmod{p}\), which means there exists an element \(b \in \mathbb{Z}_p\) such that \(b^2 \equiv -1 \pmod{p}\). So, we can write \(X^4 + 1 \equiv (X^2 - b^2 + 2bX)(X^2 - b^2 - 2bX) \pmod{p}\), which is reducible.

Step 5: Conclude the proof

In all cases, we find that \(X^4 + 1\) is reducible in \(\mathbb{Z}_p[X]\) for all prime numbers \(p\). This concludes the proof.

Key Concepts

Prime numbers

Prime numbers are the building blocks of number theory. A prime number is any natural number greater than 1, which has no positive divisors other than 1 and itself. Here are some key points about prime numbers:

• They are essential in various fields such as cryptography, coding theory, and computer science.
• Examples include 2, 3, 5, 7, etc.
• Any number greater than 1 is either a prime or can be factored into prime numbers.

Understanding prime numbers is crucial when analyzing polynomial behavior over different fields, as the conditions change significantly when dealing with prime moduli.

Fermat's Little Theorem

Fermat's Little Theorem is a fundamental result in number theory that provides insight into the properties of numbers in modular arithmetic. It states that if we have a prime number \(p\) and an integer \(a\) such that the greatest common divisor of \(a\) and \(p\) is 1, then:

• \(a^{p-1} \equiv 1 \pmod{p}\)

This theorem is particularly useful for finding roots of polynomials modulo a prime. In the context of the polynomial \(X^4 + 1\), it offers a tool to simplify equations and test reducibility. By raising elements to powers like \(p-1\), we can explore potential solutions and factor polynomials by understanding how elements behave under modulo \(p\).

Complex numbers

Complex numbers add a fascinating layer to solving polynomial equations. A complex number is a number of the form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit satisfying \(i^2 = -1\). Important aspects include:

• Complex numbers expand the real number system, allowing the solution of equations that have no real solutions.
• The roots of polynomials like \(X^4 + 1\) can be explored in the complex plane by expressing them in terms of \(i\).
• Understanding complex roots can help factor polynomials over different fields.

Using complex numbers, we can describe polynomial zeros and their behavior in quadratic and higher-degree polynomials, essential for determining reducibility over specific fields.