# Normal basis theorem

Let be a (finite-dimensional) Galois extension of a field . Then there exists a normal basis for , that is, a basis consisting of an orbit of the Galois group . Thus, an element generates a normal basis if and only if its conjugates , , are linearly independent over ; see, e.g., [a3]. The element is called a normal basis generator or a free element in . A far-reaching strengthening of the normal basis theorem is due to D. Blessenohl and K. Johnsen [a1]: There exists an element that is simultaneously free in for every intermediate field .

Such an element is called completely free (or completely normal). For the important special case where is a Galois field, a constructive treatment of normal bases and completely free elements can be found in [a2].

#### References

[a1] | D. Blessenohl, K. Johnsen, "Eine Verschärfung des Satzes von der Normalbasis" J. Algebra , 103 (1986) pp. 141–159 |

[a2] | D. Hachenberger, "Finite fields: Normal bases and completely free elements" , Kluwer Acad. Publ. (1997) |

[a3] | N. Jacobson, "Basic algebra" , I , Freeman (1985) (Edition: Second) |

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Normal basis theorem.

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