Gauss' Quadratic Reciprocity Law Could someone explain simply, the significance of Quadratic Reciprocity? I recall it being referred to as a ``crowning achievement,'' but I failed to grasp why. 
 A: It gives an extremely powerful and completely unexpected relationship between different prime numbers. Recall that for two different primes $p$ and $q$, both congruent to 1 modulo 4 for simplicity, it states that
$$\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right). $$
In words, this is saying that $p$ is a square modulo $q$ if and only if $q$ is a square modulo $p$. The more you think about it, the stranger this seems - how do things modulo $p$ 'know about' things modulo $q$? They're different primes. In fact, knowing things like the Chinese Remainder Theorem may lead to one to say things like 'nonsense, this theorem can't possibly be true, because the behaviour of integers modulo distinct primes is independent'.
And this is, indeed, pretty much true when dealing with linear equations. When we step up to quadratics, however, something completely unexpected happens, and it turns out there is a relationship.
Try this out. Take $p=5$ and $q=97$. Is $97$ a square modulo $5$? This is easy to find out, since we can immediately notice that $97\equiv 2\pmod{5}$ and then just calculate the squares modulo $5$, which are $0,\pm 1$. So $97$ is not a square modulo $5$. Quadratic reciprocity then tells us that $5$ is also not a square modulo $97$. This is much less obvious! Naively one would have to go through all the squares modulo 97 and see if any are congruent to 5. Quadratic reciprocity lets you just check things modulo 5 instead.
So it's completely bizarre that, even though numbers behave independently modulo different primes in a linear sense, when talking about squares, there is this very precise relationship. This hints at a deeper structure, which allows one to prove all kinds of things about numbers, and is the birth of modern number theory.  
