It's nice to know that you have genuine interest in math. Let me give you a brief overview of what math is.
I don't what it means to say that math is the only "constant" thing in the universe. If by this you mean, math is the only "true" thing in the universe, you might want to define "truth." Math is true only with respect to the current accepted axiomatic framework. Some axioms that mathematicians accept are the ZFC axioms of set theory and the Peano axioms. Now, any axiomatic system with arithmetic (of natural numbers), has inherent limitations, in that some mathematical statements about natural numbers can't be proven to be true nor proven to be false. Such statements are said to be undecidable. Here's an example of an undecidable statement:
Take any natural number n. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. Repeat the process and you'll eventually reach 1.
Let me talk about algebra, and the theory behind when polynomials are solvable by radicals.
You know that a polynomial with degree 2 is clearly solvable by radicals, since the roots are given by the quadratic formula. The question of when a polynomial of any degree (with rational coefficients) is solvable by radicals can be answered using group theory.
Definition: A group is a set G with a binary operation * that satisfies the following properties:
For any x,y,z in G
- x*y is in G
- x*(y*z) = (x*y)*z
-There exists an identity element e in G such that e*x = x*e = x
-Every element has an inverse element b such that x*b = b*x = e
Notice that I haven't mentioned commutativity. If the operation satisfies commutativity then our group (G, *) is said to be an
abelian group.
Some example of groups are the integers with addition, the real numbers with addition, the rational numbers with addition, and the rational numbers without 0 with multiplication. Typically they're written as (Z, +), (R, +), (Q,+) and (Q, *) respectively.
Now we can have functions between groups that satisfy certain properties. Let G, H be groups (let's drop the operation sign for simplicity).
A function f: G --> H is said to be a homomorphism if for any x,y in G
f(xy) = f(x)f
An example of a homomorphism is f: Z --> Z defined by f(x) = x. Typically, functions like these that send every element to itself are called identity functions.
We also have what's called a subgroup of a group, which is simply a subset of a group that also forms group under the same operation. If H is a subgroup of G, then we write H < G to indicate this.
Now back to homomorphisms, we must be aware of a certain class of homomorphisms called isomorphisms. Isomorphisms are homomorphisms that are invertible (equivalently bijective homomorphisms). The identity map mentioned earlier is clearly an isomorphism.
There are special types of subgroups known as normal subgroups. For a subgroup H < G if a is in G, we let aH ={ ax : x is in H}, that is the set of all elements of H multiplied (on the left side) by a (note: if the operation of the group isn't mentioned, we just call it "multiplication"). We define Ha similarly. If it turns out that aH = Ha, H is said to a
normal subgroup of G. What's so interesting these is that they allow us to construction a new group called the quotient group of G by H denoted G/H. We define G/H as {aH : a is in G}. That is the set of cosets of H. The operations is simply aH *bH = abH. This defines a group with the identity element being H itself.
Now let's tackle the question we asked at the beginning. Given a polynomial p(x) with rational coefficients, we define a splitting field of p(x) to be the smallest extension of Q(the rational numbers), over which p(x) factors as
a(x - b1)(x - b2).....(x - bn)
For instance, the polynomial x^2 - 2 has the splitting field Q(√2) which is defined to be
{q(√2)/r(√2) : q, r are polynomials with rational coeffs and r(√2) =/= 0}
Let's introduce a couple more concepts from group theory. A group G is said to be solvable if it has subgroups G1,.....Gk such that
{1} < G1 < .... Gk < G
and each subgroup is normal in the next and Gk/Gk-1 is an abelian quotient group. Note that here, 1 = e
Let E be the splitting field of p(x). We define the galois group of E over Q , denoted Gal(E/Q) to be the set of all isomorphisms that fix Q pointwise that is
{f: E --> E : f is an isomorphism and f(x) = x for every x in Q}
This forms a group under the operation of function composition.
Theorem: A polynomial p(x) with rational coefficients is solvable by radicals if and only if Gal(E/Q) is a solvable group.
The proof is way too long to give here, but here's a nice corollary.
Corollary(Abel-Ruffini): The general polynomial of degree greater than or equal to 5 with rational coefficients (quintics or higher) is not solvable by radicals.
That's a taste of what math is.
