normal basis

English

Noun

normal basis (plural normal bases)${\displaystyle }$

1. (algebra, field theory) For a given Galois field 𝔽_q^m and a suitable element β, a basis that has the form {β, β^q, β^q2, ... , β^qm-1}.

   A normal basis is generated by the repeated action of the Frobenius endomorphism on a suitable element β; it is the orbit of β for that endomorphism.

   It is a characterising property of normal bases that $\beta ^{q^{m}}=\beta$.

   • 1989, Willi Geiselmann, Dieter Gollmann, Symmetry and Duality in Normal Basis Multiplication, T. Mora (editor), Applied Algebra, Algebraic Algorithms, and Error-correcting Codes: 6th International Conference, Proceedings, Springer, LNCS 357, page 230,

     We also combine dual basis and normal basis techniques. The duality of normal bases is shown to be equivalent to the symmetry of the logic array of the serial input / parallel output architectures proposed in this paper.

   • 2006, Falko Lorenz, translated by Silvio Levy, Algebra: Volume I: Fields and Galois Theory, Springer, page 260:

     12.3 In the finite field $E=\mathbb {F} _{3^{3}}$, find: (a) a primitive root of $E$ whose conjugates do not form a normal basis of $E/\mathbb {F} _{3}$; (b) a normal basis that does not consist of primitive roots of $E$.
     For an arbitrary field $E$ with prime field $\mathbb {F} _{p}$, the extension $E/\mathbb {F} _{p}$ does always have at least one normal basis consisting of primitive roots.

   • 2015, Sergey Abrahamyan, Melsik Kyureghyan, New recursive construction of normal polynomials over finite fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott (editors, Topics in Finite Fields, American Mathematical Society, page 1,

     The set of conjugates of normal element is called normal basis. A monic irreducible polynomial $F\in \mathbb {F} _{q}[x]$ is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over $\mathbb {F} _{q}$. The minimal polynomial of an element in a normal basis $\{\alpha ,\alpha ^{q},\dots ,\alpha ^{q^{n-1}}\}$ is $\textstyle m(x)=\prod _{i=0}^{n-1}(x-\alpha ^{q^{i}})\in \mathbb {F} _{q}[x]$ which is irreducible over $\mathbb {F} _{q}$. The elements of a normal basis are exactly the roots of some N-polynomial. Hence an N-polynomial is just another way of describing a normal basis.

Derived terms

• normal basis theorem
• primitive normal basis

Translations

particular type of basis of a finite field

• French: base normale m

Further reading

• Basis (linear algebra) on Wikipedia.Wikipedia
• Polynomial basis on Wikipedia.Wikipedia
• Zech's logarithm on Wikipedia.Wikipedia
• normal basis on Wikipedia.Wikipedia

Anagrams

• isabnormals