To activate Latex support in this blog, I have now activated the QuickLatex plugin. This has solved the problems with vertical alignment of mathematics that previously plagued this blog (only in new posts using QuickLatex, the back catalogue has not been converted).
The plugin webpage linked above shows its usage (for both posts and comments), allowing Latex to be typed natively once the word “latexpage” in square parentheses appears (so no inserting of the word “latex” after a dollar symbol anymore).
be the binary tetrahedral group. This group appears as the double cover of the group of rotations of the tetrahedron (under 
), as a group of units in an appropriate 
-form of the quaternion group, or as 
.
. Now consider the Galois group of a finite Galois extension. It has a large pro-
.
are finite and number and have all been computed, together with their Galois groups. They can be found 
.
acts on 
preserving the usual pairing 
where the 
are the usual basis vectors.
, the span of 
and an invariant hyperplane 
. Let 
. 
.
on 
, the pairing 
.
is not in the kernel, since we know all normal subgroups of 
, 
and 
for A, B and C respectively. Let G be 
. It is easy then to check that the midpoint D of AB is 
and that A, D and G are concurrent.
is a map of the form 
where 
is an invertible matrix and 
is a vector. The affine transformations act transitively on the set of (nondegenerate) triangles and the property of having concurrent medians is clearly invariant under these transformations.
is a unique factorisation domain).
where 
are distinct primes. Consider the finite cyclic group 
. It has a composition series where the group 
appears 
times as a simple subquotient (and no other simple factors appear). Therefore by the 
together with their multiplicities 
be a polynomial and define
. Let the set of Galois conjugates of 
be 
Then we have 
, expand this as ![\sum_{J\subset[k]}e^{\sum_{j\in J}a_j}=0](https://s0.wp.com/latex.php?latex=%5Csum_%7BJ%5Csubset%5Bk%5D%7De%5E%7B%5Csum_%7Bj%5Cin+J%7Da_j%7D%3D0&bg=ffffff&fg=000&s=0&c=20201002)
and rewrite as
are the nonzero exponents.
be an integer such that 
is an algebraic integer. Let 
be a (large) prime and we choose to take![\displaystyle f(x)=N^{dp}x^{p-1}\prod_{i=1}^d(x-\theta_i)^p\in\mathbb{Z}[x].](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28x%29%3DN%5E%7Bdp%7Dx%5E%7Bp-1%7D%5Cprod_%7Bi%3D1%7D%5Ed%28x-%5Ctheta_i%29%5Ep%5Cin%5Cmathbb%7BZ%7D%5Bx%5D.+&bg=ffffff&fg=000&s=0&c=20201002)
(independent of 
and apply the naive estimate).
. It is a Galois-invariant algebraic integer, hence an integer. We have
, hence by 
, so we get the lower bound
by 
to produce a quotient and remainder, the code is.
by fractional linear transformations:
. The LHS lies in 
and the RHS lies in 
. Therefore 
and the rest is an easy induction.
for both an element of the Weyl group and the corresponding point in 
. Let 
be a Schubert variety. Then 
is an inclusion of 
representations. The latter is generated by the 
-weight space, where 
is the highest root.
, where 
is the reflection corresponding to 
connecting 
and 
. Since this generates 
, we have 
.
, then