Tagged Questions
-5
votes
1answer
119 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
45 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
72 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
45 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 ...
0
votes
2answers
54 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
41 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
68 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 ...
3
votes
2answers
203 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
58 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
97 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 ...
2
votes
1answer
53 views
Find the Mobius transformation mapping $(i, 0, \infty)$ to $(0, \infty, -i)$, in precisely that order
I did this:
Assuming my Mobius 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
72 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
139 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
63 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
125 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
137 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
355 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 $\dfrac {n} {abc\ldots k}$, ...
0
votes
2answers
74 views
Wikipedia plot of $\deg(\mathrm{minpoly})$ of complex numbers?
Regarding this picture on Wikipedia article for Algebraic numbers. The description is:
Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate degree of ...
13
votes
1answer
355 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
614 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
169 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
727 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
121 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
154 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
1k 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 ...
