# Tagged Questions

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### Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
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### 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 ...
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### Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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### Abstract Algebra Group theory

Let $G:= \{ e^z \;:\; z\in\mathbb{C}\}$ form a group under multiplication. Question : G is isomorphic with which group?
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### 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 ...
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### 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 ...
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### 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 $\mathbb{R}$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\pm i$ ...
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### 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 ...
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### 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 $(a,b)(c,d)=(ac-db,ad+bc)$ ...
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### 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.
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### 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 ...
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### $\mathbb{Z}[\sqrt{-23}]$: 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$is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $1/z$
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### 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 $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
### $x^2+1=0$ 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 ...