Intereting Posts

Show that $e^x > 1 + x + x^2/2! + \cdots + x^k/k!$ for $n \geq 0$, $x > 0$ by induction
Distributing 6 oranges, 1 apple, 1 banana and 1 pineapple among 3 children
Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$
marginal probability mass functions
Proving a sequence defined by a recurrence relation converges
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
Dominated convergence theorem with $f_n(x)=\frac{1}{\sqrt{2\pi}}e^{-\sqrt{n}x}\left(1+\frac{x}{\sqrt{n}}\right)^n\chi_{}$
Prove that a compact metric space is complete.
Countable family of Hilbert spaces is complete
A Curious Binomial Sum Identity without Calculus of Finite Differences
Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?
Book suggestion for probability theory
Number theory question with automorphic numbers
Two exercises on characters on Marcus (Part 2)
Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent

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 $\mathbb{R}^{+}$ of positive real numbers under multiplication.

- Let $R$ be a commutative unital ring. Is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?
- Ring of formal power series over a principal ideal domain is a unique factorisation domain
- Rings of integers are noetherian (question about a specific proof)
- Classification of fields which are isomorphic to some finite extension
- What is a projective ideal?
- Find a single-valued analytic branch of $\sqrt{z^2-1}$ in $\mathbb{C} \backslash $.
- What is the “standard basis” for fields of complex numbers?
- Galois field extension and number of intermediate fields
- What does it mean for something to hold “up to isomorphism”?
- Sum of unit and nilpotent element in a noncommutative ring.

Define $f:{\mathbb C}^{\times}\to T\times\mathbb{R}^{+}$ by $f(z)=(\frac{z}{|z|},|z|)$ and prove that this is an isomorphism of groups.

Actually $\mathbb{C}^{\times}$ is the *internal* direct product of the two since ${\mathbb C}^{\times}=T\mathbb{R}^{+}$ (write $z=\frac{z}{|z|}\cdot|z|$) and $T\cap\mathbb{R}^{+}=\{1\}$.

Hint: take the following “other-direction” map as that of YACP:

$$g: \Bbb T\times \Bbb R^+\to \Bbb C^*\;;\;\;g(e^{it},r):=re^{it}\;,\;\;t\in\Bbb R$$

i.e., the polar-coordinates representation for a non-zero complex number.

- Proving Hölder's Inequality
- Do Convergence in Distribution and Convergence of the Variances determine the Variance of the Limit?
- Definition of e
- Perfect Square relationship with no solutions
- Mapping Irregular Quadrilateral to a Rectangle
- Local diffeomorphism is diffeomorphism provided one-to-one.
- Why is the 2 norm “special”?
- Reasoning the calculation of the Hilbert's distance
- Does this condition on the sum of a function and its integral imply that the function goes to 0?
- Group with order $p^2$ must be abelian . How to prove that?
- Applications of Gauss sums
- $f_n(x_n)\to f(x) $ implies $f$ continuous – a question about the proof
- Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$
- Lower and upper bounds for the length of phi-chains wanted
- Definition of conditional expectation of a random variable given another one