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Chapter 6: Problem 21

(a) Let \(a\) be a non-zero integer, \(p\) a prime, \(n\) a positive integer, and \(p \times n\). Prove that \(p \mid \Phi_{n}(a)\) if and only if \(a\) has period \(n\) in \((\mathbf{Z} / p \mathbf{Z})^{*} .\) (b) Again assume \(p X n\) Prove that \(p \mid \Phi_{n}(a)\) for some \(a \in \mathbf{Z}\) if and only if \(p=1\) mod \(n\). Deduce from this that there are infinitely many primes \(\equiv 1\) mod \(n\), a special case of Dirichlet's theorem for the existence of primes in an arithmetic progression.

Short Answer

Expert verified
In summary, we have shown that \(p \mid \Phi_n(a)\) if and only if \(a\) has period \(n\) in \((\mathbf{Z} / p \mathbf{Z})^*\). Moreover, we proved that \(p \mid \Phi_n(a)\) for some \(a \in \mathbf{Z}\) if and only if \(p \equiv 1\) mod \(n\). Using Dirichlet's theorem, we deduced that there are infinitely many primes that are congruent to \(1\) mod \(n\).

Step by step solution

01

(a) Prove: p | 桅_n(a) iff a has period n in (Z/pZ)鈭�

First of all, let's define the notations we are working with. Let \(a\) be a non-zero integer, \(p\) a prime, and \(n\) a positive integer. We know that \(p\) divides \(n\). The Euler's totient function is denoted by \(\Phi_n(a)\). And the period of an integer \(a\) modulo \(p\), denoted by \(ord_p(a)\), is the smallest positive integer \(n\) such that \(a^n \equiv 1\) mod \(p\). Now, recall that \(\Phi_n(a) = a^n - 1\). Let's rewrite the expression \(\Phi_n(a)\) as: \[ \Phi_n(a) = a^n - 1 = (a^n - 1) \mod p \] We will now consider two cases. 1) Let \(p \mid \Phi_n(a)\). Then, \(a^n \equiv 1\) mod \(p\). The period of \(a\) in \((\mathbf{Z} / p \mathbf{Z})^*\) is the smallest integer \(n\) for which \(a^n \equiv 1\) mod \(p\). Since \(a^n \equiv 1\) mod \(p\) and \(p \mid \Phi_n(a)\), we can say \(a\) has the period \(n\) in \((\mathbf{Z} / p \mathbf{Z})^*\). 2) Suppose \(a\) has period \(n\) in \((\mathbf{Z} / p \mathbf{Z})^*\). By definition, it means that \(a^n \equiv 1\) mod \(p\). Thus, \(\Phi_n(a) = a^n - 1 \equiv 0\) mod \(p\). In this case, \(p \mid \Phi_n(a)\). So, we have shown that \(p \mid \Phi_n(a)\) if and only if \(a\) has period \(n\) in \((\mathbf{Z} / p \mathbf{Z})^*\).
02

(b) Prove: p | 桅_n(a) for some a in Z iff p = 1 mod n

We need to prove that \(p \mid \Phi_n(a)\) for some \(a \in \mathbf{Z}\) if and only if \(p \equiv 1\) mod \(n\). By definition, \(p \mid \Phi_n(a)\) implies \(a^n \equiv 1\) mod \(p\). By Fermat's Little Theorem, we know that \(a^{p-1} \equiv 1\) mod \(p\) for any \(a\) not divisible by \(p\). If \(a^n \equiv 1\) mod \(p\) for some \(a\), then \(p-1\) must be divisible by \(n\). This happens if and only if \(p \equiv 1\) mod \(n\). To show there are infinitely many primes congruent to \(1\) mod \(n\), notice that \(p \equiv 1\) mod \(n\) implies \(p \mid \Phi_n(a)\) for some \(a \in \mathbf{Z}\). Now, we can consider \(+1\), \(-1\) units in the ring of integers modulo \(n\), and construct an arithmetic progression of the form \(1+in\), for \(i=1, 2, 3, ...\), with common difference equal to \(n\). Thus, there is an infinite sequence of numbers that are \(\equiv 1\) mod \(n\). By Dirichlet's theorem for the existence of primes in an arithmetic progression, there are infinitely many primes that are \(\equiv 1\) mod \(n\). Therefore, there are infinitely many primes \(p\) for which \(p \equiv 1\) mod \(n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Number
A prime number is a fundamental concept in number theory and mathematics as a whole. A prime is an integer greater than 1 that is not the product of two smaller natural numbers. In other words, a prime number is divisible only by 1 and itself.

For example, the numbers 2, 3, 5, and 7 are prime because they cannot be divided evenly by any other numbers (except for 1 and the number itself). On the other hand, the number 6 is not prime because it is the product of 2 and 3.

Prime numbers are essential in various fields, from cryptography to the study of prime distributions. They carry a special significance in mathematical problems and theorems, including Euler's totient function and Dirichlet's theorem on arithmetic progressions, which involve the existence of primes within certain numerical patterns.
Arithmetic Progression
An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant. This constant is often referred to as the 'common difference'. If the first term of the sequence is 'a' and the common difference is 'd', then the sequence is: \[a, a + d, a + 2d, a + 3d, \dots\]

An example of an arithmetic progression is the series 3, 5, 7, 9, ..., where the common difference is 2. Arithmetic progressions are not only simple patterns; they also have a variety of applications across different areas of mathematics and are deeply connected to the study of primes, as demonstrated by Dirichlet's theorem.
Dirichlet's Theorem
Dirichlet's theorem on arithmetic progressions has a significant role in number theory. It states that for any two positive coprime integers 'a' and 'd', there are infinitely many primes of the form 'a + nd', where 'n' is a non-negative integer.

This theorem guarantees that there are endless prime numbers in sequences that might not immediately appear to contain them. For instance, in the sequences 2, 5, 8, 11, ... or 3, 7, 11, 15, ..., as long as the first term and the common difference are coprime, we can be sure that there are primes within this pattern that come up infinitely often.

Connecting to the Original Problem

In the context of the original problem, Dirichlet's theorem underlies the reasoning that leads to the conclusion that there are infinitely many primes congruent to 1 mod 'n'. This deep understanding of arithmetic progressions and prime distributions allows mathematicians to make powerful statements about the nature of prime numbers and their occurrence within well-structured numerical sets.

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Most popular questions from this chapter

Let \(k\) be a field and \(X\) a variable over \(k\). Let $$ \varphi(X)=\frac{f(X)}{g(X)} $$ be a rational function in \(k(X)\), expressed as a quotient of two polynomials \(f, g\) which are relatively prime. Define the degree of \(\varphi\) to be max(deg \(f\), deg \(g\) ). Let \(Y=\varphi(X)\). (a) Show that the degree of \(\varphi\) is equal to the degree of the field extension \(k(X)\) over \(k(Y)\) (assuming \(Y \notin k\) ). (b) Show that every automorphism of \(k(X)\) over \(k\) can be represented by a rational function \(\varphi\) of degree 1 , and is therefore induced by a map $$ X_{\mapsto} \cdot \frac{a X+b}{c X+d} $$ with \(a, b, c, d \in k\) and \(a d-b c \neq 0 .\) (c) Let \(G\) be the group of automorphisms of \(k(X)\) over \(k\). Show that \(G\) is generated by the following automorphisms: \(t_{b}: X \mapsto X+b, \quad \sigma_{a}: X \mapsto a X \quad(a \neq 0), \quad X \mapsto X^{-1}\) with \(a, b \in k\).

Let \(k\) be a field of characteristic \(\neq 2 .\) Let \(c \in k, c \notin k^{2} .\) Let \(F=k(\sqrt{c}) .\) Let \(\alpha=a+b \sqrt{c}\) with \(a, b \in k\) and not both \(a, b=0\). Let \(E=F(\sqrt{\alpha})\). Prove that the following conditions are equivalent. (1) \(E\) is Galois over \(k\). (2) \(E=F\left(\sqrt{\alpha^{\prime}}\right)\), where \(\alpha^{\prime}=a-b \sqrt{c}\). (3) Either \(\alpha \alpha^{\prime}=a^{2}-c b^{2} \in k^{2}\) or \(c \alpha \alpha^{\prime} \in k^{2}\). Show that when these conditions are satisfied, then \(E\) is cyclic over \(k\) of degree 4 if and only if \(c \alpha \alpha^{\prime} \in k^{2} .\)

Let \(k\) be a field of characteristic \(p\), and consider \(W(k) .\) Then \(V\) is an additive endomorphism of \(W(k)\), and \(F\) is a ring homomorphism of \(W(k)\) into itself. Furthermore, if \(x \in W(k)\) then $$ p x=V F x $$ If \(x, y \in W(k)\), then \(\left(V^{i} x\right)\left(V^{\prime} y\right)=V^{i+j}\left(F^{D i} x \cdot F^{N j} y\right) .\) For \(a \in k\) denote by \(\\{a\\}\) the Witt vector \((a, 0,0, \ldots)\). Then we can write symbolically $$ x=\sum_{i=0}^{\infty} V^{i}\left\\{x_{i}\right\\} $$ Show that if \(x \in W(k)\) and \(x_{0} \neq 0\) then \(x\) is a unit in \(W(k)\). Hint: One has $$ 1-x\left\\{x_{0}^{-1}\right\\}=V y $$ and then $$ x\left\\{x_{0}^{-1}\right\\} \sum_{0}^{\infty}(V y)^{i}=(1-V y) \sum_{0}^{\infty}(V y)^{i}=1 $$

Let \(k\) be a field such that every finite extension is cyclic, and having one extension of degree \(n\) for each integer \(n .\) Show that the Galois group \(G=G\left(k^{a} / k\right)\) is the inverse limit \(\lim \mathbf{Z} / m \mathbf{Z}\), as \(m \mathbf{Z}\) ranges over all ideals of \(\mathbf{Z}\), ordered by inclusion. Show that this limit is isomorphic to the direct product of the limits $$ \prod_{p} \lim _{n \rightarrow \infty} \mathbf{Z} / p^{n} \mathbf{Z}=\prod_{p} \mathbf{Z}_{p} $$ taken over all prime numbers \(p\), in other words, it is isomorphic to the product of all p-adic integers.

Prove that there are infinitely many non-zero integers \(a, b \neq 0\) such that \(-4 a^{3}-27 b^{2}\) is a square in \(\mathbf{Z}\).
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