February 18, 2009
At the end of last lecture we arrived at the question of whether every finite field is a for some prime
In this lecture we show that this is not the case, by exhibiting a field of 4 elements. We also find some general properties of finite fields. Finite fields have many interesting applications (in cryptography, for example), but we will not deal much with them as our focus through the course is on number fields, that we will begin discussing next lecture.
We begin by proving the following result:
Lemma 13. Suppose that is a finite field. Then there is some natural number such that the sum of ones vanishes, The least such is a prime that divides the size of the field.
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