...In the same vein, we get the following result about cardinalities of finite fields. 3.11 Corollary: The number of elements in a finite field is a power of some prime. Proof: Let Kbs a finite field. In Section 1 we saw that the characteristic of K must be...

.... . + anutn, where a, e Z/pZ. It follows that F has p" elements. We have proved Proposition 7.1.3. The number of elements in a finite field is a power of a prime. Ife is the identity of the finite field F, let p be the smallest integer such that pe = 0. We have...

...or prime characteristics. Wedderburn's Theorem. Every finite division ring is commutative (a field). The number of elements in a finite field is a power of its characteristics (see 7.15). Any two finite fields with the same number of elements are isomorphic....

...shall need. For the proof of these properties see Van der Waerden [ I ] vol. 1 , Chapter V, §37. 1) The number of elements in a finite field is a power of a prime. For each prime power p, there is a finite field GF(p,] with р, elements, and it is unique to within...