Figuring out whether a ring is a field Given a ring, how do you test whether it is a field? What properties would you look at?
 A: 
A field is a nonzero commutative ring that contains a multiplicative
  inverse for every nonzero element

This is from http://en.wikipedia.org/wiki/Field_%28mathematics%29
A: I'm going to assume you mean a commutative ring with a multiplicative unit. There are a few ways you can check that such a ring is a field.


*

*(perhaps the most common) Find an explicit inversion operation, i.e. a function $i : F \to F$ such that $x \cdot i(x) = 1$ for every nonzero $x\in F$.

*If the ring is finite, you need only prove it has no zero divisors, because this means that multiplication by nonzero constants is injective, which in the finite case means that it is surjective (so bijective, so has an inverse).

*Alternatively, prove that there are no nontrivial ideals. If every ideal is trivial then all the principal ideals, i.e. the sets $\{ xa : a \in F \}$, contain $1$ (or are zero). Hence every nonzero $x$ has some $a$ such that $xa = 1$, so $a$ is an inverse for $x$.

*

*A related idea would be to prove that your ring is the quotient of some other ring by a maximal ideal.


