TABLE OE CONTENTS, 



FIEST PAKT. 



INTRODUCTION TO THE GALOIS FIELD THEORY. 



CHAPTER I. 



Definition and properties of finite fields. 



Section 



1 3. Classes of residues with respect to a prime modulus . . . 3 4 



4. Format's theorem 4 



5. Definition of a field 5 



67. Definition of a Galois Field 68 



8 10. Order of a finite field is a power of a prime 9 10 



1117. Period of a mark of a field; primitive roots 1112 



18. Every finite field may be represented as a Galois Field . . 13 14 



CHAPTER II. 



Proof of the existence of the GF[pm] for every prime p 

 and integer m. 



19 22. Decomposition of functions belonging to the GF[p n ] . . 14 15 



23 25. Irreducible factors of x p ' x 15 16 



26 27. Expression for product of all irreducible quantics of degree m 



in the GF[pn]. Their number 1718 



Exercices ' 19 



CHAPTER HI. 

 Classification and determination of irreducible quantics. 



29 30. Exponent to which an irreducible quantic belongs .... 19 20 



31 32. Roots of an irreducible quantic; their exponents 21 



33. When ^ + aj*-i-j M-fl is irreducible . . 21 



34 38. Determination of irreducible quantics in the GF[pn] whose 



degree contains no prime factor other than those of pn 1 . 22 27 



3946. Irreducible quantics of degree p* in the GF[pn] 2832 



47 49. Miscellaneous theorems on irreducible quantics 32 34 



50 58. Primitive roots and primitive irreducible quantics .... 35 42 



59. Exercises 4244 



60. Table, of primitive irreducible quantics 44 



