# Tag: abstract-algebra

## abstract algebra – Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)?

The Question : 108 people think this question is useful Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions (“triernions”). Yet no one uses these. Why is this? The Question Comments : For a perfect analogy with the relevant

## abstract algebra – How to find the Galois group of a polynomial?

The Question : 105 people think this question is useful I’ve been learning about Galois theory recently on my own, and I’ve been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the technical stuff. A specific example would be how to

## abstract algebra – Why are the solutions of polynomial equations so unconstrained over the quaternions?

The Question : 117 people think this question is useful An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$, but in $\mathbb H$, ${\bf i}\cos x + {\bf j}\sin x$

## abstract algebra – Why “characteristic zero” and not “infinite characteristic”?

The Question : 114 people think this question is useful The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we define it to be $0$. But why characteristic zero? Why do

## linear algebra – Is there an “inverted” dot product?

The Question : 113 people think this question is useful The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$ What about the quantity? $$\mathbf{a} \star \mathbf{b} = \prod_{i=1}^{n} (a_{i} + b_{i}) = (a_{1} +b_{1})\,(a_{2}+b_{2})\cdots \,(a_{n}+b_{n})$$ Does it have a name? “Dot sum” seems largely inappropriate. Come to think

## abstract algebra – When to learn category theory?

The Question : 111 people think this question is useful I’m a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I’ve also had a short course on number theory which used some basic concepts about groups and modular arithmetic. Is it too early to

## abstract algebra – More than 99% of groups of order less than 2000 are of order 1024?

The Question : 111 people think this question is useful In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? Is there a nice way or

## abstract algebra – Classification of prime ideals of $\mathbb{Z}[X]$

The Question : 119 people think this question is useful Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? $(0)$. $(f(X))$, where $f(X)$ is an irreducible polynomial. $(p)$, where $p$ is a

## abstract algebra – The square roots of different primes are linearly independent over the field of rationals

The Question : 138 people think this question is useful I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I’ve tried to solve the problem using elementary algebra and also using the theory of field extensions, without

## abstract algebra – Does $R[x] \cong S[x]$ imply $R \cong S$?

The Question : 132 people think this question is useful This is a very simple question but I believe it’s nontrivial. I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic. Thanks! If there isn’t

## abstract algebra – Example of infinite field of characteristic $p\neq 0$

The Question : 129 people think this question is useful Can you give me an example of infinite field of characteristic $p\neq0$? Thanks. The Question Comments : The Answer 1 187 people think this answer is useful One very important example of an infinite field of characteristic $p$ is $$\mathbb{F}_p(T)=\left\{\,\frac{f}{g}\,\Bigg|\,\,\,f,g\in\mathbb{F}_p[T], g\neq0\right\},$$ the rational functions in

## terminology – What are the differences between rings, groups, and fields?

The Question : 127 people think this question is useful Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used? The Question Comments : Is there a diagram somewhere that depicts the relationships pictorially? Ah, this page contains some useful diagrams concerning group