Prove that any Algebraic Closed Field is Infinite
Prove that any algebraic closed field is infinite.
Definition.
A field $F$ is said to be algebraically closed if each non-constant polynomial in $F[x]$ has a root in $F$.
Proof.
Let $F$ be a finite field and consider the polynomial
\[f(x)=1+\prod_{a\in […]
Successful Probability of a Communication Network Diagram
Consider the network diagram in the figure. The diagram consists of five links and each of them fails to communicate with probability $p$. Answer the following questions about this network.
(1) Determine the probability that there exists at least one path from A to B where every […]
Upper Bound of the Variance When a Random Variable is Bounded
Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$.
\[V(X) \leq \frac{c^2}{4}.\]
Proof.
Recall that […]
Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
Proof.
The group $G$ acts on the set of left cosets $G/H$ by left multiplication.
Hence […]
Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$
Let $T: \R^3 \to \R^2$ be a linear transformation such that
\[T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
4
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
2 \\
5
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
3 \\
6 […]
Determine a 2-Digit Number Satisfying Two Conditions
A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45.
Find the number.
Solution.
The key to this problem is noticing that our 2-digit number can be […]