Math Forum / Mathematics / Undergraduate Math / December 2008

One of the more remarkable things I have is a weirdly simple result on
the complex plane which brings into question Galois Theory, which is
just such a huge thing that it's hard to surmount disbelief despite
the simplicity of the proof.

In the ring of algebraic integers consider the special construction:
7*(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)
where the a's are roots of
a^2 - (7x-1)a + (49x^2 - 14x) = 0

Hello, I was just reading you're link about slope and couldn't figure out w=
ho invented it.=A0 Also do you know why "m" is used in the mathematical equ=
ation?
Thanks,

In the ring of algebraic integers consider the special construction:
7*(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)
where the a's are roots of
a^2 - (7x-1)a + (49x^2 - 14x) = 0

I have a set of points to which I want to fit a sphere. I have a
little minimizer-based fitting algorithm that works when all the
points are all on one side of the sphere (e.g., the points' elevations
are all above 60 degrees or so). However, the fitting process is less

As I was trying to prove (or find a counterexample for) a proposition
relating to sub-groups, I realized that every example of a sub-group
(of a finite group) that I came up with was cyclic.
I don't know if this is because I have limited imagination, or if

let g be any connected graph and v any vertex of g. show that it is
possible to place arrows on the edges of G so that every vertex of the
graph is reachable from v
any ideas?

Prove that the number of people who have ever lived on earth, and who
have shaken hands an odd number of times, is even.
I know we use the theorem # of odd vertices in any graph has to be
even......right?

It seems that I haven't convinced with my talk of the distributive
property so here is an example which should test your understanding of
Galois Theory to the limit.
In the ring of algebraic integers, let x=1+sqrt(-6), with

Before I came up with my special construction any one of you if
presented with the following would call it easily:
7*P(x) = (f_1(x) + 7)(f_2(x) + 2), where f_1(0) = f_2(0) = 0
I'm sure you'd call it trivial as well that f_1(x) is the product of

Well when we talk about differentiation we say that d/dx is the
differential operator and acts on a function of x say but when we move
onto solving differential equations why do we split dy and dx say for
solving a simple seperable equation.

Could someone tell me the proper or the most common way to pronounce
these terms:
Homogeneous
Orthonormal

Any ideas how to solve these
1. A tree has the following degree sequence: 5,4,3,2,1,1...,1  How
many leaves does it have?
2. Prove that a tree with a vertex of degree d must have at least d

I've got an ellipse of the form:
z (x, y coordinate of the centre)
a and b (its axis)
alpha (rotation angle)

So I have an extremely easy proof of this extraordinary somewhat
subtle error that entered the mathematical field in the late 1800's,
which I can dramatically prove with a simple construction:
7(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)