1
vote
2answers
79 views

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 ...
1
vote
1answer
52 views

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 ...
1
vote
1answer
65 views

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 ...
1
vote
2answers
61 views

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 ...
2
votes
2answers
63 views

About Path-connected

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 ...
1
vote
0answers
45 views

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 ...
2
votes
1answer
41 views

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 ...
2
votes
0answers
34 views

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 ...
0
votes
1answer
48 views

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 $f: V \times V \rightarrow \mathbb{C}$ ...
2
votes
1answer
47 views

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 ...
6
votes
1answer
349 views

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 ...
5
votes
1answer
98 views

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 ...
4
votes
2answers
238 views

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 ...
2
votes
1answer
154 views

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 ...
11
votes
1answer
241 views

“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 ...
1
vote
3answers
255 views

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 ...
0
votes
1answer
140 views

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 $x<y$ ...
0
votes
1answer
59 views

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 $\mathbb{C} \cup \{\infty\}$, ...
2
votes
0answers
65 views

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 ...
-2
votes
1answer
90 views

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 ...
2
votes
2answers
202 views

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 $\mathbb{C}.$
1
vote
1answer
47 views

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 ...
0
votes
1answer
42 views

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 ...
11
votes
5answers
665 views

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 ...
11
votes
1answer
465 views

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 ...
4
votes
0answers
188 views

Möbius transformations form a simple group

How to show the group $M$ of Möbius 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 ...
1
vote
1answer
199 views

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 ...
4
votes
2answers
177 views

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!
4
votes
4answers
239 views

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 ...
1
vote
2answers
144 views

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 Möbius transformation on the extended comlex plane. What I ...
0
votes
1answer
46 views

Let $M_{\alpha}, \alpha \in \mathbb{C}$ be the subgroup of the Möbius transformation mapping $\alpha$ to itself. Calculate $M_i$

Let $M_{\alpha}, \alpha \in \mathbb{C}$, be the subgroup of $M$ mapping $\alpha$ to itself, that is, the stabilizer of $\alpha$. Given that $$M_0 = \left \{w = \frac{z}{cz + d}, d \neq 0 ...
1
vote
1answer
91 views

Why the kernel of isogeny is finite?

It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact. I have some questions about this. 1. Following is my reason for the kernel is discrete. ...
0
votes
2answers
130 views

Definition of an affine Möbius transformation

If I have define a Möbius transformation as "a map on the extended complex plane, $\bar{\mathbb{C}} \rightarrow \bar{\mathbb{C}}$, given by $\omega = \frac{az + b}{cz + d}$ where $a,b,c,d \in ...
3
votes
1answer
37 views

Compute the subgroup $M_{\{0, -3, \infty\}} \subset M$ consisting of 6 transformations, preserving the set $\{0, -3, \infty\}$

Compute the subgroup $M_{\{0, -3, \infty\}} \subset M$ consisting of 6 transformations, preserving the set $\{0, -3, \infty\}$, together with an explicit isomorphism $$ M_{\{0, -3, \infty\}} = S_3 $$ ...
5
votes
6answers
392 views

The notion of complex numbers

How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will ...
0
votes
0answers
64 views

Why do most special functions not have a multiple zero?

Why do most special functions not have a multiple zero ? Let $z$ be a complex number. Let $f(z)$ be a special function meromorphic in the entire complex plane. Then the zero's $x_i$ such that $f(x_i) ...
0
votes
1answer
234 views

There is a simple explanation that shows why the Fundamental Theorem of Algebra can not be proved without results of Analysis?

A "challenge" that graduate students often do in Algebra for students doing a first course in algebra is: "Prove the Fundamental Theorem of Algebra without using the results of analysis." To study ...
1
vote
2answers
734 views

Find the order of the cyclic subgroup of the given group generated by the indicated element

$\mathbf{29.}$ The subgroup of $U_6$ generated by $\cos\frac{2\pi}3+i\sin\frac{2\pi}3.$ $\mathbf{30.}$ The subgroup of $U_5$ generated by $\cos\frac{4\pi}5+i\sin\frac{4\pi}5.$ $\mathbf{31.}$ The ...
2
votes
1answer
516 views

$n$-sheeted branched covering

Michael Artin's algebra let $f(x,y)$ be an irreducible polynomial in $\mathbb{C}[x,y]$ which has degree $ n>0$ in the variable $y$. The Riemann surface of $f(x,y)$ is an $n$-sheeted branched ...
255
votes
7answers
7k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
3
votes
3answers
341 views

Is there a “homomorphism/isomorphism” (conceptually) between $\mathbb{C - R}$ and $\mathbb{R - Q}$

The Background: I was thinking about ways to conceptualize the way rational and irrational numbers interact within the real line while helping somebody in an elementary real analysis class. It came to ...
5
votes
1answer
289 views

Towards the solution of the Problem : Field Extension problem beyond $\mathbb C$ (Question 1)

I am posting this problem in order to break the problem in my previous post Field Extension problem beyond $\mathbb C$. Notation: $M(\mathbb C):=$ Field of all meromorphic functions on $\mathbb C$, ...
2
votes
1answer
303 views

Show that $\alpha$ is a zero of multiplicity $k.$

Show that $\alpha$ is a zero of multiplicity $k$ if and only if $$P(\alpha)=P'(\alpha)=\cdots =P^{(k-1)}(\alpha)=0$$ and $P^{(k)}(\alpha)\neq 0.$ So by definition of multiplicity, ...
3
votes
1answer
568 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
1answer
96 views

Convert Arrays of Reals into an Equivalent Array of Complex

Given a collection of $n$ real number arrays of length $m$, for example: $$[r_{11},\ \dots, r_{1m}]$$ $$\vdots$$ $$[r_{n1},\ \dots, r_{nm}]$$ is it possible to transform the entire collection into ...
4
votes
1answer
110 views

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 $Z(P)$ ...
5
votes
1answer
95 views

Maximal ideals of an algebra

How do I find the maximal ideals of the algebra of holomorphic functions in one variable?thanks.
2
votes
0answers
311 views

Preparing for reading Penrose's “Road to Reality”

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 ...
4
votes
2answers
156 views

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, ...
3
votes
1answer
556 views

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 ...