# Tagged Questions

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### Ring of rational-coefficient power series defining entire functions

I'm wondering if anyone has come across the following ring before. Let $R$ be the ring of complex power series $f=\sum_{n \ge 0} a_n t^n$ such that $a_n \in \mathbb{Q} \: \: \forall \: n$ The ...
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### Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
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### alternate proof of the fundamental theorem of algebra

I was reading over my notes from complex analysis and saw the fundamental theorem of algebra which states that: A polynomial of positive degree over a field $\mathbb{C}$ of complex numbers has a ...
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### Complex analysis - existence of field $\mathbb{C}$

In the following: $\mathbb{F}$ is defined to be a field containing $\mathbb{R}$ and in which the equation $x^{2}+1=0$ can be solved. Then define a set $\mathbb{C}$ to be subset of $\mathbb{F}$ whose ...
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Let $a=(a_1,a_2,...,a_k)$ and $b=(b_1,b_2,...,b_k)$ be points in k-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0,1]$ with values in ...
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### Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
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### How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider ...
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### Fractional linear transformations and the extended complex plane in a more abstract context?

Does anyone know of an "abstract algebra-esque" treatment of the extended complex plane and the Mobius transformations? I am studying complex analysis now, and I am a little frustrated that my ...
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### Sesquilinear forms seen as bilinear maps

Let $V$ be a complex vector space. A sesquilinear map (or conjugate-linear in the first variable and linear in the second) on a complex vector space $V$ is a map $f: V \times V \rightarrow \mathbb{C}$ ...
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### Is the given Ring $\mathcal K(U)$ an integral domain?

Let $U$ be a bounded open disk in $\mathbb C$ and $\mathcal K(U)$ denote the ring of complex analytic function on $U$. Is $\mathcal K(U)$ an integral domain . Give an example of a maximal ideal in ...
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### Why is the polynomial $S(\vec{x})$ with coefficients obeying a constraint homogeneous?

I have recently been working on a problem to prove that a particular polynomial is in fact homogeneous. Although I have found out that this is true, I am curious to see whether there might be a deeper ...
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### Is there an algebraic invariant for complex curves that's mapped to injectively?

Consider the functor $\pi_1: \text{Closed Surfaces} \rightarrow \textbf{Grp}$. This is homotopy invariant; every closed topological surface has a unique fundamental group. In the reverse direction, by ...
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### homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.

Let $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ be a homomorphism of the multiplicative group of complex numbers to the multiplicative group of real numbers. I need to show that the kernel of $f$ must ...
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### Differentiability vs Analyticity

What makes the crucial difference between the reals and the complex numbers is that the complex numbers are algebraically closed. So while going through all the proofs that "being holomorphic implies ...
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### “Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
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### Roadway and book recommendations to math study.

I had some calculus, linear algebra and complex analysis courses back in college. But it is not comprehensive. And I felt that my college math was not taught in a logical sequence (maybe because my ...
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### Automorphism of $\mathbb R$ fixing $\mathbb Q$ is identity

I will first show the work that I have done and I will then explain what I have left to prove. I have shown that for $\sigma\in Aut_{\mathbb{Q}}(\mathbb{R})$, then if $x,y\in\mathbb{R}$, with $x<y$ ...
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### Differentiable functions on closed and open sets in $\mathbb{C}$

Is there a difference between functions holomorphic (on open sets $\Omega$) and functions that have derivatives everywhere on $\mathcal{Cl}(\Omega)$ (their closure in $\mathbb{C} \cup \{\infty\}$, ...
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### Solving a system of rational functions

Given $c_k \in \mathbb C$ distinct nonzero complex numbers. Is that possible to find a nontrivial solution for $\vec x = (x_1, x_2, \dots, x_n) \in \mathbb C^n$ such that $\vec x$ satisfies the ...
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### Logic and its connection [closed]

My main interests are Algebra,Complex (fourier/laplace) analysis and Differential equations. In the future i plan to know a lot of abstract algebra/linear algebra/universal algebra/functional ...
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### Diagonalizability in $\mathbb{R}$ and $\mathbb{C}$

Give an example of a matrix $A\in M_{n\times n}(\mathbb{R})$ that is not diagonalizable, but A is diagonalizable viewed as a matrix over the field of complex numbers $\mathbb{C}.$
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### Uniqueness theorem for Rational Functions

I know that for polynomials $P,Q$, the equation $P(z) \equiv Q(z)$ implies that they are of the same degree and have the same coefficients. Is there an analogous result for rational fucntions? That ...
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### Injections of subfields of $\mathbb{C}$ into $\mathbb{C}$

I am having trouble understanding the last step in the proof of this theorem:"To every $w$ in $\mathcal{F}_{1}$ corresponds a rational function $r$, with coefficients in $\mathcal{F}_{2}$, such that ...
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### Difference between $\mathbb C$ and $\mathbb R^2$

What are the basic differences between $\mathbb C$ and $\mathbb R^2$? The points in these two sets are written as ordered pairs, I mean the structure looks similar to me. So what is the reason to ...
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### Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
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### Möbius transformations form a simple group

How to show the group $M$ of MÃ¶bius transformations is a simple group? I know: $SL_2(\mathbb C)/\{+I,-I\}\cong M$ then if $A \lhd M \implies \phi^{-1}(A) \lhd SL_2(\mathbb C)/\{+I,-I\}$. So if ...
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### The winding number of a curve and the fundamental theorem of algebra

One of my textbooks argues the plausibility of the fundamental theorem of algebra by using the fact (unknown to me) that the winding number of curve in the complex plane undergoing continuous ...
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### How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!
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### Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
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### Möbius transformation: Compute the subgroup $M_R \subset M$, preserving the line $R = \{x = 0\} \subset \mathbb{R}^2 = \mathbb{C}$

Compute the subgroup $M_R \subset M$, preserving the line $$R = \{x = 0\} \subset \mathbb{R}^2 = \mathbb{C}.$$ Here $M$ is a MÃ¶bius transformation on the extended comlex plane. What I ...
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### The zero set of sums of polynomials

As I am new to this forum, please correct me if this post is not appropriate. In that case I apologize. Let $P(z)$ and $Q(z)$ be polynomials with coefficients in $\mathbb{C}$, furthermore let $Z(P)$ ...
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### Maximal ideals of an algebra

How do I find the maximal ideals of the algebra of holomorphic functions in one variable?thanks.
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I am reading Road to Reality by Roger Penrose and I although I know about calculus, complex analysis, differential equations I do not know about manifolds, Riemann surfaces and so on. Which books can ...
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### Subrings of formal series rings

Let $k$ be a field and $A = k[[x_1, \dots, x_n ]]$ be the ring of formal series in $n$ variables. Consider $g_1, \dots, g_m \in A$ such that $g_1(0) = \cdots = g_m(0) = 0$. For every \$f \in k[[t_1, ...
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### Polynomials, Rouche's theorem and index of vector fields

In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two ...