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 […]

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 […]

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 $\textit{identity element}$, $F \ne \emptyset$. […]

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? 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 […]

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 […]

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 $\mathbb{Z}[x]$.

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 […]

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 […]

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.)

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