Articles of abstract algebra

Is this quotient Ring Isomorphic to the Complex Numbers

So the question goes: Let $$ A=\mathbb{R}[x]/\langle x^2-x+1\rangle . $$ Is A isomorphic to $ \mathbb{C} $ ? The earlier parts of the question asked for me to a) find the reciprocal in A of $x+1+I$ and b) find $p(x)+I\in A$ such that $(p(x)+I)^2=-1+I$. I found the answers to both of these parts, but I […]

When $\Bbb Z_n$ is a domain. Counterexample to $ab \equiv 0 \Rightarrow a\equiv 0$ or $b\equiv 0\pmod n$

Suppose $$ab \equiv 0 \mod n$$ and that $a$ and $b$ are positive integers both less than $ n$. Does it follow that either $a | n$ or $b | n$? If it does follow, give a proof. If it doesn’t, then give an example. For this question is this a suitable answer: No it […]

Fields – Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies that any parentheses are meaningless. Therefore, we will not use parentheses. Therefore, we will not use $\textit{associativity}$ explicitly. By $\textit{identity element}$, $F \ne \emptyset$. […]

What are the differences between rings, groups, and fields?

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?

Can $G≅H$ and $G≇H$ in two different views?

Can $G≅H$ and $G≇H$ in two different views? We have two isomorphic groups $G$ and $H$, so $G≅H$ as groups and suppose that they act on a same finite set, say $\Omega$. Can we see $G≇H$ as permutation groups. Honestly, I am intrested in this point in the following link. It is started by: Notice […]

Transpose inverse automorphism is not inner

Consider the transpose inverse automorphism on $GL_n(\mathbb F)$ where $n\geq2$ and $|\mathbb F|>2$. (i.e. $\mathbb F$ is a field, possibly infinite, with three or more elements). I want to show this automorphism is not inner. I was told to consider $\det(BAB^{-1}) = \det(A)$ and $\det(\,^TA^{-1})=\det(A)^{-1}$ and derive a contradiction. However, in some fields (where the […]

Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z $

Hi I don’t know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$.

Proving a commutative ring can be embedded in any quotient ring.

Here’s the exercise, as quoted from B.L. van der Waerden’s Algebra, Show that any commutative ring $\mathfrak{R}$ (with or without a zero divisor) can be embedded in a ”quotient ring” consisting of all quotients $a/b$, with $b$ not a divisor of zero. More generally, $b$ may range over any set $\mathfrak{M}$ of non-divisors of zero […]

System of linear equations having a real solution has also a rational solution.

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $b ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $Ax = b$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution in $\mathbb{Q}^n$? and I thought I’d give an interesting, possibly wrong, approach to solving it. I’m not sure if such things […]

$\mathbb{Q}/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$

How can I prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain? Also, I need to prove that its field of fractions is isomorphic to the field of rational functions $\mathbb{Q}(t)$? (The question is taken from UC Berkeley Preliminary Exam, Fall 1995.)