Problem 11 - Algebra

Question

Prove that the field \(\mathbf{Q}_{p}\) contains all \((p-1)\) -th roots of unity. [Hint : Use Proposition 7.6, applied to the polynomial \(X^{p-1}-1\) which splits into factors of degree 1 in the residue class field. ] Show that two distinct \((p-1)\) -th roots of unity cannot be congruent mod \(p\).

Step-by-Step Solution

Answer

Using Proposition 7.6, we see that the polynomial \(f(X) = X^{p-1}-1\) in the field \(\mathbf{Q}_{p}\) has \(p-1\) distinct roots, which are the \((p-1)\)-th roots of unity. We also showed that no two distinct \((p-1)\)-th roots of unity can be congruent modulo \(p\). Thus, \(\mathbf{Q}_{p}\) contains all \((p-1)\)-th roots of unity.

Step 1: State Proposition 7.6

Proposition 7.6 states that if a polynomial \(f(X)\) with coefficients in a field \(K_p\) has exactly \(n\) distinct roots in \(K_p\) (counted with multiplicity), then its reduction \(\overline{f}(X)\) in the residue field \(\overline{K_p}\) is the product of linear factors with the same multiplicity.

Step 2: Apply Proposition 7.6 to the polynomial \(X^{p-1}-1\)

We apply Proposition 7.6 to the polynomial \(f(X) = X^{p-1}-1\) with coefficients in the field \(\mathbf{Q}_{p}\). Note that the reduction of \(f(X)\) in the residue field is \(\overline{f}(X) = X^{p-1}-1\). By Proposition 7.6, this polynomial splits into factors of degree 1 in the residue class field, which means that there are \(p-1\) distinct roots in \(\mathbf{Q}_{p}\). Each of these roots, say \(\zeta_i\), is a \((p-1)\)-th root of unity, i.e., satisfies the equation \(\zeta_i^{p-1} = 1\).

Step 3: Show that two distinct \((p-1)\)-th roots of unity cannot be congruent mod \(p\)

Suppose for contradiction that there exist two distinct \((p-1)\)-th roots of unity \(\zeta_a\) and \(\zeta_b\); where \(\zeta_a \neq \zeta_b\), and they are congruent modulo \(p\). This implies that \(\zeta_a \equiv \zeta_b \pmod{p}\), or equivalently, \(\zeta_a - \zeta_b \equiv 0 \pmod{p}\). Now, we raise both sides of the equation to the power of \(p-1\), \((\zeta_a - \zeta_b)^{p-1} \equiv 0^{p-1} \pmod{p}\). By Fermat's Little Theorem, we know that \(a^{p-1} \equiv 1 \pmod{p}\) for any non-zero integer \(a\). Since \(\zeta_a^{p-1} = \zeta_b^{p-1} = 1\), we get \((1 - 1)^{p-1} \equiv 0 \pmod{p}\), which contradicts Fermat's Little Theorem, as we do not expect \(0^{p-1} \equiv 0 \pmod{p}\) when \(p-1 > 1\) (counting multiplicative inverses). Thus, no two \((p-1)\)-th roots of unity can be congruent modulo \(p\). In conclusion, we have shown that the field \(\mathbf{Q}_{p}\) contains all \((p-1)\)-th roots of unity and no two distinct roots can be congruent modulo \(p\).

Key Concepts

Polynomial Factorization

Understanding polynomial factorization is an essential part of algebra that deals with breaking down a complex polynomial into simpler, more manageable parts (factors). This concept plays a significant role in finding the roots of polynomials, which are the solutions where the polynomial equals zero.

In the context of the presented exercise, the polynomial in question is of the form X^{p-1} - 1, where p is a prime number. This particular polynomial is special because it's linked to the concept of roots of unity; these are complex numbers that, when raised to a certain power (in this case, p-1), equal one.

When we factorize X^{p-1} - 1 over the field \(\textbf{Q}_p\), which consists of rational numbers with a prime denominator p, we are able to find p-1 distinct linear factors. Each factor corresponds to a different (p-1)-th root of unity within the field. This demonstrates one of the most fascinating aspects of polynomial factorization: its ability to unveil the underlying structure and symmetries within a mathematical field.

Fermat's Little Theorem

Fermat's Little Theorem is a fundamental theorem in number theory that states for any integer a and prime p, if p does not divide a, then a^{p-1} \equiv 1 \pmod{p}. This powerful theorem helps us understand the nature of numbers within modular arithmetic and has applications in cryptography, prime testing, and more.

In relation to our exercise, Fermat's Little Theorem justifies why two distinct (p-1)-th roots of unity, \(\textbf{\zeta_a}\) and \(\textbf{\zeta_b}\), cannot be congruent modulo p. If they were, we could then derive a contradiction to Fermat's Little Theorem, which tells us that the expression (\(\textbf{\zeta_a}\) - \(\textbf{\zeta_b}\))^{p-1} should be equivalent to 1 \pmod{p} and not zero, as the assumption of congruence would require. This contradiction highlights the theorem's utility in proving properties about the structure of fields and the behavior of their elements.

Residue Class Field

The concept of a residue class field is tied to the idea of congruence in modular arithmetic. The field is made up of equivalence classes, with two numbers being equivalent if they differ by a multiple of a given modulus, typically a prime p. Every element in a residue class field can be represented as an integer from 0 to p-1, and these elements behave similarly to integers under addition and multiplication modulo p.

This relates to our study of roots of unity within the field \(\textbf{Q}_p\), as it implies that the polynomial X^{p-1} - 1 breaks down into linear factors in the residue class field formed by the modulus p. The proposition used in the exercise demonstrates that this factorization reveals each (p-1)th root of unity as an element of the field, ensuring their uniqueness and existence within the field. As such, the residue class field is a key concept for understanding the behavior of polynomials and their roots in the context of modular arithmetic.