Articles of abstract algebra

Prove that G is a group

The exercise is: Let $g\in G$. $G$ is a group. Prove that $G=\{gx:x\in G$}. I know the the definition of group but the proof that is in the book is the next one: Let $H=\{gx:x\in G\}$ $H\subseteq G$ because $g\in G$ and $x\in G$ $\Rightarrow gx \in G$ G $\subseteq H$ (?) (by definition) Let […]

Determine invariant subspaces

imagine that a matrix of an endomorphism has the characteristic polynomial $(\lambda-2)^2(\lambda-3)$ now i was wondering whether all invariant subspaces can be determined by $0,V$ and $\ker(A-2)^2, \ker(A-2), \ker(A-3)$? or how do I find them?

Classical Impossible Constructions of Geometry proofs with Abstract Algebra.

Trisecting an angle (dividing a given angle into three equal angles), Squaring a circle (constructing a square with the same area as a given circle), and Doubling a cube (constructing a cube with twice the volume of a given cube). Told that these problems could only be proved with abstract algebra. I have no idea […]

Notation for modules.

Given a module $A$ for a group $G$, and a subgroup $P\leq G$ with unipotent radical $U$, I have encountered the notation $[A,U]$ in a paper. Is this a standard module-theoretic notation, and if so, what does it mean. In the specific case I am looking at, it works out that $[A,U]$ is equal to […]

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if $(s-1)+i$ is even.) My attempt: If $s=2^m$, then $s$ is even. This means $s-1$ is […]

Simple ring and field

We know that the center of a simple ring with unity is a field. But I couldn’t make an example of a ring which is not simple but its center is a field. Is it possible? Please give a hint.

Analog of Kronecker's Theorem for Polynomials over an Algebra

The usual is stated something like this: Kronecker’s Theorem: if $p(x) \in \mathbb{F}[x]$ is a monic $n$-th order irreducible polynomial where $\mathbb{F}$ is a field then $\mathbb{F}[x]/(p(x))$ forms a degree $n$ extension field of $\mathbb{F}$. Moreover, if $\theta = x \ mod(p(x))$ then $\mathbb{F}[x]/(p(x))$ has basis $1, \theta, \dots , \theta^{n-1}$. For example, $\mathbb{R}[x]/(x^2+1) \approxeq […]

Quaternion algebra and norm

Let $a \in \mathbb{Q}$ be a nonzero rational number and set $(5,a)$ and (for the associated division algebras over $\mathbb{Q}$). Let us suppose that $b$ is the norm of some element of $\mathbb{Q}[\sqrt{5}]$. How can I write down an explicit isomorphism between $(5,a)$ and $(5,ba)$?

Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal

Let $f:R\to S$ be a surjective homomorphism, where $R$ is a commutative ring and $S$ is a field. Prove that $\ker(f)$ is a maximal ideal. I already know that $\ker(f)$ is an ideal of $R$. I tried to consider some ideal $J$ of $R$ such that $\ker(f) \subset J$. If we can show that for […]

Characteristics subgroups, normal subgroups and the commutator.

Let $G$ be a group and let $N$ be a normal subgroup of $G.$ Let $N’$ denote the commutator of $N.$ Prove that $N’$ is a normal subgroup of $G.$ What I do know is that the commutator subgroup is characteristics. What I am not sure about is whether or not $N’$ is characteristic in […]