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Normal Basis

A normal basis of a finite Galois extension field L/K is a vector basis consisting of the conjugates of one element under the Galois group. The normal basis theorem states that every finite Galois extension field has such a basis (Milne 2022, p. 68).

For a finite field extension F_(q^n)/F_q, a normal basis has the form

{alpha,alpha^q,...,alpha^(q^(n-1))}.

For example, if alpha^2+alpha+1=0 in F_4, then {alpha,alpha^2} is a normal basis over F_2. An element that generates a normal basis over every intermediate field is a completely normal element.

See also

Completely Normal Element, Galois Extension Field, Vector Basis

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References

Lidl, R. and Niederreiter, H. Finite Fields, 2nd ed. Cambridge, England: Cambridge University Press, 1997.Milne, J. S. Fields and Galois Theory, Version 5.10. Sep. 2022. https://www.jmilne.org/math/CourseNotes/FT.pdf.

Cite this as:

Weisstein, Eric W. "Normal Basis." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NormalBasis.html

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