Finitely Fine

Number Theory Level pending

Is it possible to have a field (or set) of numbers such that you can do every basic operation to it (addition, subtraction, multiplication, division) but the number of elements is finite?

For example, the set of rationals is a field, but it is infinite. is it possible to have a field that is finitely big?

Yes No

Jacob Day by Jacob Day , 17, USA

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1 solution

Jacob Day
Sep 25, 2017

Yes! It is possible within a certain modulus, such as mod 3 or mod 5! For example, {0, 1, 2, 3, 4} is a finite set called Z sub 5. Addition is supported, for example, 4 + 2 = 6 which in mod 5 is 1. So is subtraction, multiplication, and division.

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In fact, for any prime p, the numbers in mod p work. (I believe there're ways to construct finite fields with size that're powers of primes, but I'm not as familiar with the exact mechanics of how to do it. mod p^k doesn't form a field, since any multiple of p won't have a multiplicative inverse in it. (ie, in, say, mod 17^3, you wouldn't be able to divide by 5*17 = 85)

Psy Kosh - 3 years, 3 months ago

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