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• A One-Line Proof that there are Infinitely Many Prime Numbers Prove that there are infinitely many prime numbers in ONE-LINE.   Background There are several proofs of the fact that there are infinitely many prime numbers. Proofs by Euclid and Euler are very popular. In this post, I would like to introduce an elegant one-line […]
• The Number of Elements in a Finite Field is a Power of a Prime Number Let $\F$ be a finite field of characteristic $p$. Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$. Proof. First note that since $\F$ is a finite field, the characteristic of $\F$ must be a prime number $p$. Then $\F$ contains the […]
• Three Equivalent Conditions for an Ideal is Prime in a PID Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent. (1) The ideal $(a)$ generated by $a$ is maximal. (2) The ideal $(a)$ is prime. (3) The element $a$ is irreducible. Proof. (1) $\implies$ […]
• Every Prime Ideal of a Finite Commutative Ring is Maximal Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$. Proof. We give two proofs. The first proof uses a result of a previous problem. The second proof is self-contained. Proof 1. Let $I$ be a prime ideal […]
• Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57 Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group. Then determine the number of elements in $G$ of order $3$.   Proof. Observe the prime factorization $57=3\cdot 19$. Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$. By […]
• Normal Subgroup Whose Order is Relatively Prime to Its Index Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Suppose that the order $n$ of $N$ is relatively prime to the index $|G:N|=m$. (a) Prove that $N=\{a\in G \mid a^n=e\}$. (b) Prove that $N=\{b^m \mid b\in G\}$.   Proof. Note that as $n$ and […]
• If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian? Let $R$ be a commutative ring with $1$. Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$. Is it true that $A$ is also a Noetherian ring?   Proof. The answer is no. We give a counterexample. Let […]
• The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$ Suppose that $p$ is a prime number greater than $3$. Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$. (a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$. (b) Determine the index $[G : S]$. (c) Assume […]