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The Primitive Normal Basis Theorem – Without a Computer
Author(s) -
Cohen Stephen D.,
Huczynska Sophie
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003782
Subject(s) - primitive element , normal basis , mathematics , multiplicative group , basis (linear algebra) , element (criminal law) , multiplicative function , combinatorics , extension (predicate logic) , generator (circuit theory) , galois group , prime (order theory) , finite field , primitive root modulo n , order (exchange) , discrete mathematics , group (periodic table) , power (physics) , mathematical analysis , geometry , physics , computer science , programming language , finance , quantum mechanics , political science , law , economics
Given q , a power of a prime p , denote by F the finite field GF( q ) of order q , and, for a given positive integer n , by E its extension GF( q n ) of degree n . A primitive element of E is a generator of the cyclic group E *. Additively too, the extension E is cyclic when viewed as an FG ‐module, G being the Galois group of E over F . The classical form of this result – the normal basis theorem – is that there exists an element α ∈ E (an additive generator) whose conjugates{ α , α q , … , α q n − 1}form a basis of E over F ; α is a free element of E over F , and a basis like this is a normal basis over F . The core result linking additive and multiplicative structure is that there exists α ∈ E , simultaneously primitive and free over F . This yields a primitive normal basis over F , all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [ 5 ] (completing work by Carlitz [ 1 , 2 ] and Davenport [ 4 ]).