Intereting Posts

Topology and analytic functions
Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?
Problem when integrating $e^x / x$.
integrate $\int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$
Are there useful criterions whether a positive integer is the difference ot two positive cubes?
If there are injective homomorphisms between two groups in both directions, are they isomorphic?
Is an isometry necessarily surjective?
Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
How to show that geodesics exist for all of time in a compact manifold?
How many consecutive composite integers follow k!+1?
Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$
How many integer solutions are there to the inequality $y_1 + y_2 + y_3 + y_4 < 184$
Why the gradient of $\log{\det{X}}$ is $X^{-1}$, and where did trace tr() go??
$I$-adic completion
Largest integer that can't be represented as a non-negative linear combination of $m, n = mn – m – n$? Why?

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

Thanks in advance

- Every group of order 203 with a normal subgroup of order 7 is abelian
- Rings with a given number of (prime, maximal) ideals
- Primes in a Power series ring
- The Center of $\operatorname{GL}(n,k)$
- If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
- Product of principal ideals: $(a)\cdot (b) = (a b)$

- Intuition for idempotents, orthogonal idempotents?
- Commutative integral domain with d.c.c. is a field
- Examples of categories where epimorphism does not have a right inverse, not surjective
- Why do elements of coprime orders commute in nilpotent groups?
- Possible orders of normal subgroups using only the elements in $G$ and their orders
- Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
- Number of $60$th primitive roots of $-1$
- Intersecting maximal ideals of $k$ with $k$
- How does the determinant change with respect to a base change?
- why the column sums of character table are integers?

**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 $G_{\mathbf Q}$).

- what does “the conjugacy part of sylow's theorems” denote?
- Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
- On fifth powers $x_1^5+x_2^5+\dots = y_1^5+y_2^5+\dots$
- Prove that $R \otimes_R M \cong M$
- What is continuity correction in statistics
- Integral $\int_0^1\sqrt{1-x^4}dx$
- Subadditivity of the $n$th root of the volume of $r$-neighborhoods of a set
- Does there exist a continuous $g(x,t)$ such that every continuous$ f(x)$ equals $g(x,t)$ for some $t$ and all $x$??
- How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?
- There is a bijection between irreducible components of the generic fiber and irreducible components passing through it.
- How to calculate gradient of $x^TAx$
- Correct way to calculate numeric derivative in discrete time?
- What are Diophantine equations REALLY?
- What are relative open sets?
- Derivative of an even function is odd and vice versa