1. Something I realized yesterday when reading Wikipedia is that the fundamental groups of the complements of two homeomorphic topological spaces need not be isomorphic. Here’s an example:
The fundamental group of the complement of a circle in R^3 is isomorphic to Z.
The fundamental group of the complement of a trefoil knot in R^3 is given by the presentation <a, b | a^2 = b^3>. Clearly this is not isomorphic to Z.
2. When I was reading Algebraic Number Fields by Gerald Janusz yesterday, I came across the following rather mundane-looking result:
If A is a field and B a domain which is integral over A, then B is a field.
I was expecting an equally mundane proof based on showing that every nonzero element b in B has a multiplicative inverse. To show that would not be very difficult: we merely take the minimal polynomial of b, switch the order of the coefficients, multiply by some constant, and show that this polynomial has 1/b as a root. None of these things is difficult.
However, we had just shown the following result:
Let A subset B be integral domains with B integral over A and A integrally closed. If P is a nonzero prime ideal of B, then P intersect A is a nonzero prime ideal of A.
Therefore we prove the result like this:
If B were not a field, there would exist a prime ideal P which is nonzero (and not equal to B). By the lemma, P intersect A would be a nonzero prime ideal. Since A is a field, P intersect A = A, so 1 is in P, an impossibility.