-1
votes
1answer
28 views

Group Theory proving [on hold]

can someone help me with this question? 1) Given a natural number n≥1, let $G_n$ be the set of complex n-th roots of $1$, i.e. $G_{n} = \{z \in \mathbb{C} :z^n = 1\}$ Prove that $G_n$ is a group ...
3
votes
3answers
55 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
38 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
53 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 ...
5
votes
6answers
1k views

How can you find the cubed roots of i?

I am trying to figure out what the three possibilities of $z$ are such that $$ z^3=i $$ but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you ...
1
vote
1answer
56 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 ...
3
votes
0answers
109 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
105 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
72 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 $x^7-12$ ...
-4
votes
2answers
57 views

Why there's 2 different ways to go multi-dimensional? [closed]

I have been wanting to ask this for some time. As I understand, the following 2 mathematics objects can both be considered as the attempt to expand 1 dimensional number to high dimensional number. ...
1
vote
1answer
80 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
64 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
39 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
120 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
116 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
53 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
127 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
127 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
385 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
105 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
144 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
177 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
95 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
90 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
88 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
247 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
49 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
155 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
101 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 $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
4
votes
2answers
159 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
63 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
86 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
200 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
152 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
175 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? ...
1
vote
1answer
147 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
510 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
126 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 ...
19
votes
2answers
896 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 ...
12
votes
2answers
655 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
10
votes
2answers
197 views

Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
10
votes
4answers
1k views

Why is it that Complex Numbers are algebraically closed?

I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
4
votes
1answer
133 views

Well-ordering of positive Gaussian integers under lexicographical ordering?

I am reading a paper by Richard Weimer called "Can the complex numbers be ordered?" and he makes the following claim. Let $G^+=\{a+bi : a,b$ are positive integers $ \}$ and let $<$ denote ...
7
votes
2answers
156 views

Invariant under transformation $i\mapsto -i$ implies real?

When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for ...
15
votes
5answers
2k views

How fundamental is the fundamental theorem of algebra?

Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial ...