Primitive element theorem

In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.

Terminology

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Let   be a field extension. An element   is a primitive element for   if   i.e. if every element of   can be written as a rational function in   with coefficients in  . If there exists such a primitive element, then   is referred to as a simple extension.

If the field extension   has primitive element   and is of finite degree  , then every element   can be written in the form

 

for unique coefficients  . That is, the set

 

is a basis for E as a vector space over F. The degree n is equal to the degree of the irreducible polynomial of α over F, the unique monic   of minimal degree with α as a root (a linear dependency of  ).

If L is a splitting field of   containing its n distinct roots  , then there are n field embeddings   defined by   and   for  , and these extend to automorphisms of L in the Galois group,  . Indeed, for an extension field with  , an element   is a primitive element if and only if   has n distinct conjugates   in some splitting field  .

Example

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If one adjoins to the rational numbers   the two irrational numbers   and   to get the extension field   of degree 4, one can show this extension is simple, meaning   for a single  . Taking  , the powers 1, α, α2, α3 can be expanded as linear combinations of 1,  ,  ,   with integer coefficients. One can solve this system of linear equations for   and   over  , to obtain   and  . This shows that α is indeed a primitive element:

 

One may also use the following more general argument.[1] The field   clearly has four field automorphisms   defined by   and   for each choice of signs. The minimal polynomial   of   must have  , so   must have at least four distinct roots  . Thus   has degree at least four, and  , but this is the degree of the entire field,  , so  .

Theorem statement

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The primitive element theorem states:

Every separable field extension of finite degree is simple.

This theorem applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q has characteristic 0 and therefore every finite extension over Q is separable.

Using the fundamental theorem of Galois theory, the former theorem immediately follows from Steinitz's theorem.

Characteristic p

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For a non-separable extension   of characteristic p, there is nevertheless a primitive element provided the degree [E : F] is p: indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime p.

When [E : F] = p2, there may not be a primitive element (in which case there are infinitely many intermediate fields by Steinitz's theorem). The simplest example is  , the field of rational functions in two indeterminates T and U over the finite field with p elements, and  . In fact, for any   in  , the Frobenius endomorphism shows that the element   lies in F , so α is a root of  , and α cannot be a primitive element (of degree p2 over F), but instead F(α) is a non-trivial intermediate field.

Proof

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Suppose first that   is infinite. By induction, it suffices to prove that any finite extension   is simple. For  , suppose   fails to be a primitive element,  . Then  , since otherwise  . Consider the minimal polynomials of   over  , respectively  , and take a splitting field   containing all roots   of   and   of  . Since  , there is another root  , and a field automorphism   which fixes   and takes  . We then have  , and:

 , and therefore  .

Since there are only finitely many possibilities for   and  , only finitely many   fail to give a primitive element  . All other values give  .

For the case where   is finite, we simply take   to be a primitive root of the finite extension field  .

History

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In his First Memoir of 1831, published in 1846,[2] Évariste Galois sketched a proof of the classical primitive element theorem in the case of a splitting field of a polynomial over the rational numbers. The gaps in his sketch could easily be filled[3] (as remarked by the referee Poisson) by exploiting a theorem[4][5] of Lagrange from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields.[5] Galois then used this theorem heavily in his development of the Galois group. Since then it has been used in the development of Galois theory and the fundamental theorem of Galois theory.

The primitive element theorem was proved in its modern form by Ernst Steinitz, in an influential article on field theory in 1910, which also contains Steinitz's theorem;[6] Steinitz called the "classical" result Theorem of the primitive elements and his modern version Theorem of the intermediate fields.

Emil Artin reformulated Galois theory in the 1930s without relying on primitive elements.[7][8]

References

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  1. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211. New York, NY: Springer New York. p. 243. doi:10.1007/978-1-4613-0041-0. ISBN 978-1-4612-6551-1.
  2. ^ Neumann, Peter M. (2011). The mathematical writings of Évariste Galois. Zürich: European Mathematical Society. ISBN 978-3-03719-104-0. OCLC 757486602.
  3. ^ Tignol, Jean-Pierre (February 2016). Galois' Theory of Algebraic Equations (2 ed.). WORLD SCIENTIFIC. p. 231. doi:10.1142/9719. ISBN 978-981-4704-69-4. OCLC 1020698655.
  4. ^ Tignol, Jean-Pierre (February 2016). Galois' Theory of Algebraic Equations (2 ed.). WORLD SCIENTIFIC. p. 135. doi:10.1142/9719. ISBN 978-981-4704-69-4. OCLC 1020698655.
  5. ^ a b Cox, David A. (2012). Galois theory (2nd ed.). Hoboken, NJ: John Wiley & Sons. p. 322. ISBN 978-1-118-21845-7. OCLC 784952441.
  6. ^ Steinitz, Ernst (1910). "Algebraische Theorie der Körper". Journal für die reine und angewandte Mathematik (in German). 1910 (137): 167–309. doi:10.1515/crll.1910.137.167. ISSN 1435-5345. S2CID 120807300.
  7. ^ Kleiner, Israel (2007). "§4.1 Galois theory". A History of Abstract Algebra. Springer. p. 64. ISBN 978-0-8176-4685-1.
  8. ^ Artin, Emil (1998). Galois theory. Arthur N. Milgram (Republication of the 1944 revised edition of the 1942 first publication by The University Notre Dame Press ed.). Mineola, N.Y.: Dover Publications. ISBN 0-486-62342-4. OCLC 38144376.
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