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Exercise problems in field theory in abstract algebra

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- Basic 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.
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 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 […]
- Polynomial $x^p-x+a$ 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 […]
- Every 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 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 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 […]
- Any 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 […]