The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)
Problem 519
Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).
Add to solve later Add to solve later The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD)
Problem 518
Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).
Add to solve later Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$
Problem 517
Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.
Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.
Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.
Add to solve later Idempotent Elements and Zero Divisors in a Ring and in an Integral Domain
Problem 516
Prove the following statements.
(a) If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.
(b) Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.
Add to solve later Top 10 Popular Math Problems in 2016-2017
It’s been a year since I started this math blog!!
More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).
I made a list of the 10 math problems on this blog that have the most views.
Can you solve all of them?
The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level)
Read solution
Add to solve later A Positive Definite Matrix Has a Unique Positive Definite Square Root
Problem 514
Prove that a positive definite matrix has a unique positive definite square root.
Add to solve later Find All the Square Roots of a Given 2 by 2 Matrix
Problem 513
Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$.
Find all the square roots of the matrix
\[A=\begin{bmatrix}
2 & 2\\
2& 2
\end{bmatrix}.\]
Add to solve later No/Infinitely Many Square Roots of 2 by 2 Matrices
Problem 512
(a) Prove that the matrix $A=\begin{bmatrix}
0 & 1\\
0& 0
\end{bmatrix}$ does not have a square root.
Namely, show that there is no complex matrix $B$ such that $B^2=A$.
(b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.
Add to solve later Each Element in a Finite Field is the Sum of Two Squares
Problem 511
Let $F$ be a finite field.
Prove that each element in the field $F$ is the sum of two squares in $F$.
Add to solve later The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic
Problem 510
Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.
Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.
Add to solve later How to Prove a Matrix is Nonsingular in 10 Seconds
Problem 509
Using the numbers appearing in
\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\]
we construct the matrix \[A=\begin{bmatrix}
3 & 14 &1592& 65358\\
97932& 38462643& 38& 32\\
7950& 2& 8841& 9716\\
939937510& 5820& 974& 9
\end{bmatrix}.\]
Prove that the matrix $A$ is nonsingular.
Add to solve later Eigenvalues of a Matrix and its Transpose are the Same
Problem 508
Let $A$ be a square matrix.
Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.
Add to solve later Any Automorphism of the Field of Real Numbers Must be the Identity Map
Problem 507
Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.
Add to solve later The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix
Problem 506
Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.
Namely, show that
\[(A^{\trans})^{-1}=(A^{-1})^{\trans}.\]
Add to solve later The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$
Problem 505
Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]
Using the formula, calculate the inverse matrix of $\begin{bmatrix}
2 & 1\\
1& 2
\end{bmatrix}$.
Add to solve later Every Diagonalizable Nilpotent Matrix is the Zero Matrix
Problem 504
Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.
Add to solve later The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain
Problem 503
Prove that the ring of integers
\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.
Add to solve later How to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix
Problem 502
Find the inverse matrix of the $3\times 3$ matrix
\[A=\begin{bmatrix}
7 & 2 & -2 \\
-6 &-1 &2 \\
6 & 2 & -1
\end{bmatrix}\]
using the Cayley-Hamilton theorem.
Add to solve later Every Ring of Order $p^2$ is Commutative
Problem 501
Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.
Then prove that $R$ is a commutative ring.
Add to solve later 10 True of False Problems about Nonsingular / Invertible Matrices
Problem 500
10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
You can take the quiz as many times as you like.
The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.
Add to solve later 
