0
votes
0answers
17 views

Subring of $\mathbb{Z}[i]$ and an infinite set $X$ such that $\exists x \forall y \in X \,\,x^2\mid y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases. Is there a subring $R$ of Gaussian ...
0
votes
1answer
40 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
0
votes
1answer
20 views

Euclidean division - For what values of a, does the polynomial g(t) get divided by f(t) in the complex ring

They want to find the values of a where g(t) can be divided by f(t). $f(t) = t^2 + it − ai$ $g(t) = t^4 + (1 − i)t^3 + (1 − 2i)t^2 − 3at − (4 + 2i)a$ Euclidean algorithm: $g(t) = f(t)q(t) + ...
11
votes
3answers
118 views

Prove that an expression is zero for all sets of distinct $a_1, \dotsc, a_n\in\mathbb{C}$

A while ago one of my professors gave the class a problem "to think about when lying on the beach." Well, I've been on the beach several times since then to no avail and my curiosity has finally ...
1
vote
3answers
133 views

The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the ...
1
vote
3answers
30 views

Related to the construction of $\Bbb C$ (generalisation)

To construct $\Bbb C$, we consider $\Bbb R^2$ endowed with the operations: $$\begin{align} (a,b) + (c,d) &:= (a+c, b+d) \\ (a,b) \cdot (c,d) &:= (ac - bd, ad+bc)\end{align} $$ then write ...
0
votes
3answers
33 views

need help explaing the complex roots of a cubic

I am trying to understand a Galois theory example and we are looking at the solutions of $x^3-2=0$. It says they are $2^\frac{1}{3},2^\frac{1}{3}\omega, \text{ and } 2^\frac{1}{3}\omega^2$. I know ...
0
votes
2answers
85 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
0
votes
2answers
50 views

Algebra - Gaussian integers

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ ...
0
votes
2answers
50 views

Show that $(x-\alpha)(x-\overline{\alpha})$ is a also a factor of $p(t)$ over the complex numbers

Here is the full question. Lots of struggles: Let $p(t)$ belong to $P(R)$. a) If $(x − \alpha)$ is a factor of $p(t)$ over the complex numbers (i.e. $p(t) = (x − \alpha)\cdot q(t)$, for ...
3
votes
3answers
82 views

Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
1
vote
2answers
43 views

Topologically dense subgroup

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
1
vote
1answer
62 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
1
vote
1answer
58 views

Difference between i and -i

Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a - bi$ does the image ...
1
vote
0answers
53 views

Can “polar numbers” be added in a sensible way?

Let $\mathbb{P}$ denote the set of all "polar numbers," by which I just mean pairs of real numbers $(r,\theta).$ Note in particular that $r$ is allowed to be negative. Then we can structure ...
4
votes
0answers
142 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
1
vote
2answers
159 views

Factor the polynomial $x^3 − 27$ using De Moivre's theorem (Please explain solution)

I was reading the book A First Course in Linear Algebra by Ken Kuttler (link to nearly identical page http://librarum.org/book/312/11) and I did not understand this part: Q: Factor the polynomial ...
1
vote
1answer
84 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
2
votes
2answers
175 views

Is Complex Numbers the biggest field? If yes, is there any easy proof to understand it?

Is the Complex Numbers the biggest field? If yes, does anyone have a "simple"/"easy to understand" proof?
1
vote
1answer
103 views

Closure of Algebraic Field to Complex Conjugation

I have an algebraic field $\mathbb Q(\gamma)$ with $\gamma$ the complex root of $X^3+X^2+X-1$, i.e., $\gamma\approx-0.771+1.115\mathrm i$. I have two closely related questions: Is $\mathbb ...
1
vote
1answer
65 views

Abstract Algebra Group theory

Let $G:= \{ e^z \;:\; z\in\mathbb{C}\}$ form a group under multiplication. Question : G is isomorphic with which group?
2
votes
1answer
46 views

A question about the $\ker$ of a particular group homomorphism, where the groups are both the non-zero complex numbers with multiplication.

The following statement is paraphrased from Linear Algebra: A Pure Mathematical Approach by Harvey E. Rose on page 2-3. If $G$ is a group of the non-zero complex numbers with multiplication, and ...
1
vote
1answer
136 views

Set of all homomorphisms from G into C*?

I don't really understand the nature of Ĝ as described in this question: For any group G, define its dual group Ĝ to be the set of all homomorphisms from G into $\mathbb{C}^*$, together with the ...
3
votes
2answers
121 views

Algebraic structure built from the set of complex numbers

Is it correct to say that $\mathbb{C}$ is both a field, and a dimension 2 vector space? The complex numbers can be the elements of the field, or they can be considered as 2 elements of $\mathbb{R}$, ...
0
votes
1answer
57 views

Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
2
votes
2answers
134 views

What Are the Relations for the Polar Hypercomplex form $a + bi + cj + dk$?

Olariu in "Complex Numbers in $N$ Dimensions" has polar hypercomplex numbers described by its generators as \begin{gather} \alpha^2 = \beta, \\ \beta^2 = 1, \\ \gamma^2 = \beta, \\ \alpha\beta ...
1
vote
4answers
134 views

Is the multiplication of two complex numbers with $|z|=1$ a complex number with modulus 1?

If we have two complex numbers $a, b \in \mathbb{C}$ such that $|a|=1$ and $|b|=1$ is $|a\cdot b|=1$ as well? I am trying to determine if the set $\left(\{z\in\mathbb{C}:|z|=1\},\cdot\right)$ is a ...
7
votes
2answers
430 views

How to teach a High school student that complex numbers cannot be totally ordered?

I once again need your precious knowledge! I am not sure which is the best pedagogic way to teach a High school student about why complex numbers cannot be totally ordered. When I was in High school ...
4
votes
1answer
108 views

Is there a name for a function whose square is an involution?

An involution is a function $f:X\to X$ such that $f\circ f=\text{id}$. Is there a name for a function $g:X\to X$ such that $f\equiv g\circ g$ is an involution? An example is multiplication by $\pm i$ ...
3
votes
1answer
31 views

Prove that $q(a_i)\in \{a_1,…, a_n\}$

Let $p(x)$ and $q(x)$ be polynomials with rational coefficients such that $p(x)$ is irreducible over $\mathbb{Q}$. Let $a_1,..., a_n\in \mathbb{C}$ be the complex roots of $p$, and suppose that ...
6
votes
5answers
147 views

Strong characterization of $\mathbb C$ with respect to $\mathbb R$

$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
-4
votes
1answer
184 views

Root of a quadratic equation that has modulus $1$

Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$. Show me the right way to solve this. Thanks in advance.
1
vote
2answers
114 views

Understanding bicomplex numbers

I found by chance, the set of Bicomplex numbers. These numbers took particularly my attention because of their similarity to my previous personal research and question. I should say that I can't ...
-3
votes
1answer
93 views

$\mathbb{Z}[\sqrt{-23}]$: A uniquely written set?

I suspect that $\mathbb{Z}[\sqrt{-23}] \implies \forall~z=\sqrt{23b+a}~e^{i\arctan{\frac{23b}{a}}},~\text{where $z$ is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
2
votes
1answer
91 views

ring isomorphism in the complex numbers

Let $f:\mathbb{C} \to \mathbb{C}$ be a ring isomorphism for which $f(x) = x$ for all $x\in \mathbb{R}$. Prove that $f$ is either the identity mapping ($\mathrm{id}:\mathbb{C} \to \mathbb{C}$) or f ...
1
vote
2answers
261 views

Prove the direct product of nonzero complex numbers under multiplication.

Let $\mathbb{C}^{\times}$ be the group of nonzero complex numbers under multiplication. Then $\mathbb{C}^{\times}$ is the direct product of the circle group $T$ of unit complex numbers and the group ...
3
votes
0answers
51 views

Subgroups of $P=\{z\in\mathbb{C} : z^{2n}=1 \;\mbox{for some}\; n\geq 0\}$

Investigate the subgroups of $P$ where $$P=\{z\in\mathbb{C} : z^{2n}=1 \;\mbox{for some}\; n\geq 0\}.$$ In particular, investigate the finitely generated subgroups and the infinite subgroups. ...
3
votes
2answers
165 views

Subgroups of the roots of unity.

Let $G=\mathbb{C}^*$ and let $\mu$ be the subgroup of roots of unity in $\mathbb{C}^*$. Show that any finitely generated subgroup of $\mu$ is cyclic. Show that $\mu$ is not finitely generated and find ...
4
votes
2answers
1k views

Prove that such an inverse is unique

Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$ $z^{-1}$ is also often written as $1/z$
1
vote
0answers
113 views

Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
5
votes
2answers
196 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
1
vote
1answer
64 views

Find the Möbius transformation mapping $(i, 0, \infty)$ to $(0, \infty, -i)$, in precisely that order

I did this: Assuming my Möbius transformation is some $\omega$ in terms of $z$, I want to work out a formula that gives me: 1) $\omega = 0$ when $z = i$ 2) $\omega = \infty$ when $z = 0$ 3) $\omega ...
1
vote
1answer
91 views

Representing roots of unity

Is there some notation in terms of $n,k$, I can use to represent the complex exponential $e^{2\pi i\frac{k}{n}}$, I find by writing the exponential out, I often make mistakes and it is timely to write ...
3
votes
1answer
215 views

Linear relations satisfied by nth root of unity

How do you characterize all the linear relations satisfied by $n$th roots of unity with real, integral and non-negative integral coefficients? Here are two examples for 3rd and 4th root: Let ...
0
votes
2answers
189 views

Finding a basis for a complex lattice given a nondivisible vector in the lattice

If I am given some lattice defined as, say $$L=\{Az_1+Bz_2\ |\ A,B \in\mathbb{Z}\}$$ and a vector $v=az_1+bz_2$ , where $\gcd(a,b)=1$, I would like to find another vector $\,w\in L\,$ such that ...
3
votes
3answers
194 views

How to find the roots of $x³-2$?

I'm trying to find the roots of $x^3 -2$, I know that one of the roots are $\sqrt[3] 2$ and $\sqrt[3] {2}e^{\frac{2\pi}{3}i}$ but I don't why. The first one is easy to find, but the another two roots? ...
2
votes
1answer
161 views

Next generation numbers

1: Discovering of negative numbers. Assume a and b are positive integers $x+a=b$ ----> if $b>a$ then $x$ is positive integer $x+a=b$ ----> if $b=a$ then $x=0$ $x+a=b$ ----> if ...
2
votes
1answer
547 views

primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
1
vote
2answers
149 views

Wikipedia plot of $\deg(\mathrm{minpoly})$ of complex numbers?

Regarding the following picture on the Wikipedia article for Algebraic numbers: The description is: Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours ...
22
votes
2answers
1k views

What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset ...