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

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 – 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