# Field-theory-eye-catch

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- Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$. Solution. We give several examples. The key facts are: An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain. An ideal $I$ of […]
- Find Values of $a$ so that Augmented Matrix Represents a Consistent System Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations. \[A= \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 2 &-1 & -2 & a^2 \\ -1 & -7 & -11 & a \end{array} \right],\] where $a$ is a real number. Determine all the […]
- Determine Null Spaces of Two Matrices Let \[A=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ -1 & -3 & -4 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ 5 & 3 & 3 \end{bmatrix}.\] Determine the null spaces of matrices $A$ and $B$. Proof. The null space of the […]
- Any Finite Group Has a Composition Series Let $G$ be a finite group. Then show that $G$ has a composition series. Proof. We prove the statement by induction on the order $|G|=n$ of the finite group. When $n=1$, this is trivial. Suppose that any finite group of order less than $n$ has a composition […]
- If the Order is an Even Perfect Number, then a Group is not Simple (a) Show that if a group $G$ has the following order, then it is not simple. $28$ $496$ $8128$ (b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple. Hint. Use Sylow's theorem. (See the post Sylow’s Theorem […]
- Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam) Let \[\begin{bmatrix} 0 & 0 & 1 \\ 1 &0 &0 \\ 0 & 1 & 0 \end{bmatrix}.\] (a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers? (b) Calculate $A^{2009}$. (Princeton University, […]
- Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$. Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
- Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator (a) Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module. Prove that the module $M$ has a nonzero annihilator. In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$. Here $r$ does not depend on […]