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 […]