# Michigan-State-University-Abstract-Algebra-eye-catch

• Pick Two Balls from a Box, What is the Probability Both are Red? There are three blue balls and two red balls in a box. When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red? Solution. Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]
• If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have $I_1+I_2=R.$ Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.   > Proof. Since $I_1+I_2=R$, there exists $a \in I_1$ […]
• Trace, Determinant, and Eigenvalue (Harvard University Exam Problem) (a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$. Find $\det(A)$. (b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$. (c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]
• If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix. Show that the matrix $A$ is diagonalizable. Hint. Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]
• Probability of Having Lung Cancer For Smokers Let $C$ be the event that a randomly chosen person has lung cancer. Let $S$ be the event of a person being a smoker. Suppose that 10% of the population has lung cancer and 20% of the population are smokers. Also, suppose that we know that 70% of all people who have lung cancer […]
• Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
• Cosine and Sine Functions are Linearly Independent Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$. Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.   Proof. Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]
• The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).   Proof. Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$. The (field) norm $N$ of an element $a+b\sqrt{5}$ is […]