A little puzzle

This came up as sub-problem during Young-Han’s group meeting today and we mulled over it for a few minutes but didn’t come up with a non-ugly answer. I’m sure, given the number of Real Mathematicians ™ who read this, that someone out there knows of an “obvious” explanation.

Suppose I give you p integers in an arbitrary order (where p is prime). While maintaining the order and using only addition, multiplication, and parenthesis, is it always possible to make an expression which evaluates to 0 mod p?

I think it’s true, but I’m sure there’s some special property of \mathbb{F}_p that I have forgotten. I guess further generalizations would include whether or not it’s possible for arbitrary p (not necessarily prime), how many elements of an arbitrary field you would need, and so on. I’d ask this on MathOverflow but… meh. It’s probably a homework problem.

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