Problem 2 - A Course in Galois Theory

Question

Show that if \(n\) is odd then \(\Phi_{2 n}(x)=\Phi_{n}(-x)\).

Step-by-Step Solution

Answer

If \(n\) is odd, symmetry of negative roots in \(x^n = -1\) implies \(\Phi_{2n}(x) = \Phi_{n}(-x)\) holds.

Step 1: Understand the Definitions

The function \( \Phi_{n}(x) \) is defined as the \(n\)-th cyclotomic polynomial. Cyclotomic polynomials are used to factorize expressions like \(x^n - 1\). Specifically, \(\Phi_n(x)\) is the factor in the unique factorization of \(x^n - 1\) into irreducible polynomials over the rational numbers.

Step 2: Recognize the Relationship Between Indices

If \(n\) is odd, then \(2n\) is twice an odd number. Recall that \(\Phi_{2n}(x) = \phi(2n)\), where \(\phi\) is the Euler's totient function, and it counts the number of integers up to \(2n\) that are coprime to \(2n\). For an odd \(n\), \(\phi(2n) = 2\phi(n)\).

Step 3: Express \( \Phi_{2n}(x)\) Using Roots of Unity

The \(2n\)-th roots of unity are solutions of \(x^{2n} = 1\). These include the \(n\)-th roots of unity and the \(n\)-th roots of \(-1\). This essentially means the roots are expressed as \(\Phi_{2n}(x) = \prod_{d \mid 2n} (x^d - 1)^{m(d,2n)}\), where \(m(d,2n)\) is the Moebius function.

Step 4: Transform \( \Phi_{2n}(x) \) to \( \Phi_n(-x)\)

For \(n\) odd, the expression \(x^{2n} = 1\) implies roots that have their negatives in \(x^n = -1\). This allowance to use negative implies that substituting \(-x\) in \(\Phi_{n}(x)\) generates an equivalent expression to \(\Phi_{2n}(x)\) due to symmetry across the real axis.

Step 5: Use Properties of Roots of Unity

The symmetry and properties of cyclotomic polynomials over even and odd indices allow identity \(\Phi_{2n}(x) = \Phi_{n}(-x)\) to hold, relying fundamentally on properties of the negative roots being solutions when \(n\) is odd.

Step 6: Write Conclusion

The transformation \(x^{2n} - 1 = (x^n - 1)(x^n + 1)\) illustrates the factorization property holding when substituted in and further manipulations showing the equality \(\Phi_{2n}(x) = \Phi_{n}(-x)\). Thus, the desired identity holds by means of symmetry across these roots.

Key Concepts

Roots of Unity

When we talk about the **roots of unity**, we refer to the numbers that, when raised to a certain integer power, result in 1. For any integer \( n \), the **n-th roots of unity** are the solutions of the equation \( x^n = 1 \). These roots are complex numbers and they can be expressed in the form \( e^{2\pi i k/n} \), where \( k = 0, 1, 2, \ldots, n-1 \).

• The set of n-th roots of unity forms a regular polygon in the complex plane, centered at the origin and with one vertex at 1.
• The principal n-th root of unity is \( e^{2\pi i /n} \).
• Roots of unity are key in understanding the factorization of polynomials like \( x^n - 1 \).

For instance, the equation \( x^{2n} = 1 \) provides us with the **2n-th roots of unity**, combining both the n-th roots and their reflections across the real axis. This property leads to insights into polynomial transformations, particularly detailing the relationship between \( \Phi_{2n}(x) \) and \( \Phi_n(-x) \) for odd \( n \).

Euler's Totient Function

**Euler's Totient Function**, denoted \( \phi(n) \), counts how many integers up to \( n \) are coprime with \( n \). This means those numbers that share no common divisors with \( n \) other than 1. It plays a pivotal role in understanding cyclotomic polynomials.

• If \( n \) is prime, \( \phi(n) = n-1 \) because all numbers less than \( n \) are coprime to \( n \).
• If \( n \) is odd, doubling \( n \) to become \( 2n \) means that \( \phi(2n) = 2\phi(n) \).

Because Euler's totient function helps determine the degree of cyclotomic polynomials \( \Phi_n(x) \), it directly affects their factorization properties. In the case when \( n \) is odd, understanding \( \phi(2n) \) as twice \( \phi(n) \) aids in proving that \( \Phi_{2n}(x)=\Phi_{n}(-x) \).

Moebius Function

The **Moebius function**, \( \mu(n) \), is an integer-valued function that is instrumental in number theory, especially in the inversion and recursion formulas. It's defined as follows:

• \( \mu(1) = 1 \)
• \( \mu(n) = (-1)^k \) if \( n \) is a product of \( k \) distinct primes
• \( \mu(n) = 0 \) if \( n \) has a squared prime factor

This function is useful in the factorization processes of cyclotomic polynomials. In the transformation \( \Phi_{2n}(x) = \prod_{d \mid 2n}(x^d-1)^{\mu(d,2n)} \), it's crucial in determining the exponentiated form of the factors related to roots of unity. By understanding how the Moebius function filters through factorizations, you gain deeper insights into why the cyclotomic polynomials transform in certain ways.

Factorization

**Factorization** is the process of breaking down an expression into a product of simpler expressions. For cyclotomic polynomials, this concerns the expression \( x^n - 1 \). It can be factorized uniquely over the integers.

• The formula for \( x^n - 1 \) is \( (x-1)(x^{n-1}+x^{n-2}+\ldots+1) \).
• Cyclotomic polynomials \( \Phi_n(x) \) further break this down into irreducible polynomials.

The factorization \( x^{2n} - 1 = (x^n - 1)(x^n + 1) \) emphasizes the need to understand how cyclotomic polynomials work. It reveals how the roots and their symmetries express through simpler polynomial roots \(x^{n}=1\) and \(x^{n}=-1\). This connects directly with the identity \( \Phi_{2n}(x) = \Phi_{n}(-x) \) by using elements from Euler's totient function, Moebius function, and roots of unity in its explanation.