element in field, expressed as multiple of elements?

Question

I'm new to the formal concept of a field, although I'm aware of the defining axioms. I've been struggling to think through the following idea:

Let F be a field. If I take an element a in F , is it always the case that I can find b and c in F such that b•c = a?

Any thoughts would be helpful, thanks :)

Edit: [Forgot to add this, but I mean b,c different from a]

Answer 1

The answer to your question is "yes" but it isn't a particularly informative question.

Just think about the rational numbers for a moment. Suppose $a=2$. Then you have infinitely many solutions to your problem: $$ 2 = 1 \times 2 = 6 \times (1/3) = (12/5) \times (5/6) = \cdots $$

You should see how to generalize this for any $a$ in any field.