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Field theory problems and solution in abstract algebra

Exercise problems in field theory in abstract algebra


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  • Basic Exercise Problems in Module TheoryBasic Exercise Problems in Module Theory Let $R$ be a ring with $1$ and $M$ be a left $R$-module. (a) Prove that $0_Rm=0_M$ for all $m \in M$. Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$. To simplify the […]
  • How to Diagonalize a Matrix. Step by Step Explanation.How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
  • Welcome to Problems in MathematicsWelcome to Problems in Mathematics Welcome to my website. I post problems and their solutions/proofs in mathematics. Most of the problems are undergraduate level mathematics. Here are several topics I cover on this website. Topics Linear Algebra Group Theory Ring Theory Field Theory, Galois Theory Module […]
  • Every Finite Group Having More than Two Elements Has a Nontrivial AutomorphismEvery Finite Group Having More than Two Elements Has a Nontrivial Automorphism Prove that every finite group having more than two elements has a nontrivial automorphism. (Michigan State University, Abstract Algebra Qualifying Exam)   Proof. Let $G$ be a finite group and $|G|> 2$. Case When $G$ is a Non-Abelian Group Let us first […]
  • A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is NormalA Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer. Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$. Then prove that $H$ is a normal subgroup of $G$. (Michigan State University, Abstract Algebra Qualifying […]
  • If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal SubgroupIf Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam)   Proof. To show that $H$ is a normal subgroup of […]
  • Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite FieldPolynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements. For any nonzero element $a\in \F_p$, prove that the polynomial \[f(x)=x^p-x+a\] is irreducible and separable over $F_p$. (Dummit and Foote "Abstract Algebra" Section 13.5 Exercise #5 on […]
  • Any Automorphism of the Field of Real Numbers Must be the Identity MapAny Automorphism of the Field of Real Numbers Must be the Identity Map Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.   Proof. We prove the problem by proving the following sequence of claims. Let $\phi:\R \to \R$ be an automorphism of the field of real numbers […]

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