Intereting Posts

Is $0! = 1$ because there is only one way to do nothing?
Show $2(x+y+z)-xyz\leq 10$ if $x^2+y^2+z^2=9$
How Do You Actually Do Your Mathematics?
Elliptic Regularity on Manifolds
Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?
Prove by induction that $n^2<n!$
An Integral involving $e^{ax} +1$ and $e^{bx} + 1$
Hartshorne's Exercise II.5.1 – Projection formula
Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity
Translation-invariant metric on locally compact group
$n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$
Generalization of a product measure
Normal subgroups of free groups: finitely generated $\implies$ finite index.
Solving functional equation $f(x)f(y) = f(x+y)$
Uniqueness existence of solutions local analytical for a PDE

How do you show that $Aut_\mathbb{Q}(\overline{\mathbb{Q}})$ is uncountable ?

Thanks in advance

- What's the difference between isomorphism and homeomorphism?
- Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers
- Seeing quotient groups
- What does “defining multiplication in quotient rings” actually mean?
- Generalized Eisenstein's Criterion over Integral Domains?
- Nice examples of groups which are not obviously groups

- Irreducibility of $X^{p-1} + \cdots + X+1$
- Ring homomorphisms from $\Bbb Q$ into a ring
- How many irreducible factors does $x^n-1$ have over finite field?
- Embedding Fields in Matrix Rings
- Why is it true that $|AB:A|=|B:A\cap B|$ even if $A$ is not normal in $AB$? (Second Isomorphism Theorem)
- Splitting of Automorphism Group
- Ring where irreducibles are primes which is not an UFD
- Existence of the least common multiple in a Unique Factorization Domain
- Why don't we study algebraic objects with more than two operations?
- Group of Order $p^2$ Isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p}$ $\times$ $\mathbb{Z}_{p}$

**Hint:** Show there is an infinite set of mutually independent roots of order $2$ all of which are complex, then for every subset $A$ of this set there is a permutation which conjugates all the roots of those in $A$.

Consider any chain of unequal subfields $\mathbb{Q}\subset K_1\subset K_2\subset\ldots$ that are all normal over $\mathbb{Q}$ and whose union is $\overline{\mathbb{Q}}$. Then we can extend any automorphism of $K_2/K_1$ in finitely many (but more than one) ways, and that will extend to any of the (finitely many but more than one) automorphisms of $K_3/K_2$, etc. So we can build automorphisms of the algebraic closure inductively, each one of which is an infinite sequence of finitely many choices, and there are uncountably many such sequences.

In general, a **profinite space** is either finite or uncountable. (A profinite space is a compact, totally disconnected, Hausdorff space. The proof that such a space is either finite or uncoutable is completely topological; see if you can come up with it!)

But $G_{\mathbf Q}$, which is profinite, cannot be finite, since there are finite Galois extensions of $\mathbf Q$ of arbitrary high degree (and their Galois groups are quotient of $G_{\mathbf Q}$).

- How is the Lagrangian related to the perturbation function?
- Show that there exists a vector $v$ such that $Av\neq 0$ but $A^2v=0$
- Proving $\pi^3 \gt 31$
- Prove that integrable implies bounded
- What is the Topology of point-wise convergence?
- Need help solving complicated integral $\int e^{-x}\cos4x\cos2x\,\mathrm dx$
- Funny double infinite sum
- Polar decomposition normal operator
- Weierstrass M-Test
- Is it faster to count to the infinite going one by one or two by two?
- Prove that $2^{10}+5^{12}$ is composite
- How to solve this quadratic congruence equation
- What are the continuous automorphisms of $\Bbb T$?
- Sum from 0 to n of $ n \choose i $?
- Show that (($¬n) \rightarrow (n \rightarrow \theta))$ is a theorem of L, whenever $n, \theta$ are propositional formulas.