# Math-Magic Tree Trick

• The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic.   The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chose the numbers $7, 5, 3, 2, […] • Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain Let$R$be an integral domain and let$S=R[t]$be the polynomial ring in$t$over$R$. Let$n$be a positive integer. Prove that the polynomial $f(x)=x^n-t$ in the ring$S[x]$is irreducible in$S[x]$. Proof. Consider the principal ideal$(t)$generated by$t$[…] • Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix $A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.$ (The Ohio State University, Linear Algebra Final Exam Problem) Solution. We compute the […] • Galois Group of the Polynomial$x^2-2$Let$\Q$be the field of rational numbers. (a) Is the polynomial$f(x)=x^2-2$separable over$\Q$? (b) Find the Galois group of$f(x)$over$\Q$. Solution. (a) The polynomial$f(x)=x^2-2$is separable over$\Q$The roots of the polynomial$f(x)$are$\pm […]
• Group of Invertible Matrices Over a Finite Field and its Stabilizer Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number. Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, […]
• A Matrix Representation of a Linear Transformation and Related Subspaces Let $T:\R^4 \to \R^3$ be a linear transformation defined by $T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.$ (a) Find a matrix $A$ such that […]
• The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism. If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.   Proof. We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]
• Find a Condition that a Vector be a Linear Combination Let $\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.$ Find the necessary and […]