# Stanford-university-exam-eye-catch

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- The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity. Hint. Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]
- A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator Let $R$ be a ring with $1$. A nonzero $R$-module $M$ is called irreducible if $0$ and $M$ are the only submodules of $M$. (It is also called a simple module.) (a) Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element […]
- The Inverse Matrix of an Upper Triangular Matrix with Variables Let $A$ be the following $3\times 3$ upper triangular matrix. \[A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},\] where $x, y, z$ are some real numbers. Determine whether the matrix $A$ is invertible or not. If it is invertible, then find […]
- The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns Determine all possibilities for the number of solutions of each of the system of linear equations described below. (a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution. (b) A homogeneous system of $5$ equations in $4$ unknowns and the […]
- A Linear Transformation Maps the Zero Vector to the Zero Vector Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively. Show that $T(\mathbf{0}_n)=\mathbf{0}_m$. (The Ohio State University Linear Algebra […]
- Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions. $A=aP+(a+1)Q$ $P^2=P$ $Q^2=Q$ $PQ=O$ $QP=O$, where $O$ is the $2\times 2$ zero matrix. Then do the following problems. (a) Prove that […]
- Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$ Let $G$ be a group. Suppose that we have \[(ab)^3=a^3b^3\] for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$. Then prove that $G$ is an abelian group. Proof. Let $a, b$ be arbitrary elements of the group $G$. We want […]
- Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$: \[\pi=\frac{C}{d}.\] $\pi$ in decimal starts with 3.14... and never end. I will show you several beautiful formulas for $\pi$. Art Museum of formulas for $\pi$ […]