We have already met with examples of finite fields, namely, the fields ℤ/pℤ, where p is a prime number. In this chapter we shall prove that there are many more finite fields and shall investigate their properties. This theory is beautiful and interesting in itself and, moreover, is a very useful tool in number-theoretic investigations. As an illustration of the latter point, we shall supply yet another proof of the law of quadratic reciprocity. Other applications will come later.
One more comment. Up to now the great majority of our proofs have used very few results from abstract algebra. Although nowhere in this book will we use very sophisticated results from algebra, from now on we shall assume that the reader has some familiarity with the material in a standard undergraduate course in the subject.
KeywordsFinite Field Multiplicative Group Irreducible Polynomial Finite Subgroup Principal Ideal Domain
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