Postby **blp** » Sun Oct 19, 2008 6:55 pm

I was going through some posts and noticed that many of you solve these problems in a much too complicated way. There's a very trivial theorem for verifying if something which is a subring of the complex numbers is a field. The general theorem is as follows.

Theorem: Let R be an integral domain that contains a field F. If R is finite dimensional as an F-vector space, then R is a field.

Proof: Choose an element r in R which is non-zero. Then the map R->R defined by x->rx is an F-linear map and because R is finite dimensional, it's a bijection. Hence there's an x in R that maps to 1, so we have an inverse for r. QED

The way that this can be used for anything that's contained under the complex numbers (which is the only thing you'll pretty much find on the test) is that a ring that is a subset of C is necessarily an integral domain (a subring of an integral domain is of course an integral domain). Hence something like

a+b*sqrt(3), a,b in Q

has to be an field as it has the basis (1,sqrt(3)) as a Q-vector space. Similarly something horrible like

a+b*3^(1/3)+c*3^(2/3)+d*sqrt(2)+e*sqrt(7)

is immediately seen to be a field (basis (1, 3^(1/3), 3^(2/3), sqrt(2), sqrt(7))), though it would be a horrible task to give a formula for the inverse for a general element of the ring.

EDIT: This theorem can immediately be used to confirm the affirmative of a claim. You still need to find counterexamples for the others. Usually it's just sufficient to realize that for something that's spanned only as a lattice (i.e. linear combinations of Z), you're automatically having Z itself as a subring and these elements can usually be immediately seen to have no inverse.