Primitive element theorem

In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extension is called a simple extension in this case. The theorem states that a finite extension is simple if and only if there are only finitely many intermediate fields. An older result, also often called "primitive element theorem", states that every finite separable extension is simple; it can be seen as a consequence of the former theorem. These theorems imply in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.
Terminology.
Let formula_1 be a "field extension". An element formula_2 is a "primitive element" for formula_1 if formula_4 i.e. if every element of formula_5 can be written as a rational function in formula_6 with coefficients in formula_7. If there exists such a primitive element, then formula_1 is referred to as a "simple extension". 
If the field extension formula_1 has primitive element formula_6 and is of finite degree formula_11, then every element "x" of "E" can be written uniquely in the form
where formula_13 for all "i". That is, the set
is a basis for "E" as a vector space over "F".
Example.
If one adjoins to the rational numbers formula_15 the two irrational numbers formula_16 and formula_17 to get the extension field formula_18 of degree 4, one can show this extension is simple, meaning formula_19 for a single formula_2. Taking formula_21, the powers 1, α , α2, α3 can be expanded as linear combinations of 1, formula_16, formula_17, formula_24 with integer coefficients. One can solve this system of linear equations for formula_16 and formula_17 over formula_27, to obtain formula_28 and formula_29. This shows α is indeed a primitive element: 
The theorems.
The classical primitive element theorem states: 
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.
The following primitive element theorem (Ernst Steinitz) is more general:
Using the fundamental theorem of Galois theory, the former theorem immediately follows from the latter.
Characteristic "p".
For a non-separable extension formula_1 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"] = "p"2, there may not be a primitive element (in which case there are infinitely many intermediate fields). The simplest example is formula_34, the field of rational functions in two indeterminates "T" and "U" over the finite field with "p" elements, and formula_35. In fact, for any α = "g"(T,U) in "E", the Frobenius endomorphism shows that the element "α""p" lies in "F" , so α is a root of formula_36, and α cannot be a primitive element (of degree "p"2 over "F"), but instead "F"(α) is a non-trivial intermediate field.
Constructive results.
Generally, the set of all primitive elements for a finite separable extension "E" / "F" is the complement of a finite collection of proper "F"-subspaces of "E", namely the intermediate fields. This statement says nothing in the case of finite fields, for which there is a computational theory dedicated to finding a generator of the multiplicative group of the field (a cyclic group), which is "a fortiori" a primitive element (see primitive element (finite field)). Where "F" is infinite, a pigeonhole principle proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
with "c" in "F", that fail to generate the subfield containing both elements: 
This is almost immediate as a way of showing how Steinitz' result implies the classical result, and a bound for the number of exceptional "c" in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and "a priori"). Therefore, in this case trial-and-error is a possible practical method to find primitive elements.
History.
In his First Memoir of 1831, 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 (as remarked by the referee Siméon Denis Poisson; Galois' Memoir was not published until 1846) by exploiting a theorem of Joseph-Louis Lagrange from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields. 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 two primitive element theorems were proved in their modern form by Ernst Steinitz, in an influential article on field theory in 1910; Steinitz called the "classical" one "Theorem of the primitive elements" and the other one "Theorem of the intermediate fields". Emil Artin reformulated Galois theory in the 1930s without the use of the primitive element theorems.