Add to solve later
Sponsored Links

Exercise problems in field theory in abstract algebra
Add to solve later
Sponsored Links
More from my site
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 […]