Primitive element theorem

In field theory, the primitive element theorem states that every finite separable 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

Let $E/F$ be a field extension. An element $\alpha \in E$ is a primitive element for $E/F$ if $E=F(\alpha ),$ i.e. if every element of $E$ can be written as a rational function in $\alpha$ with coefficients in $F$ . If there exists such a primitive element, then $E/F$ is referred to as a simple extension.

If the field extension $E/F$ has primitive element $\alpha$ and is of finite degree $n=[E:F]$ , then every element x of E can be written uniquely in the form

x=f_{n-1}{\alpha }^{n-1}+\cdots +f_{1}{\alpha }+f_{0},

where $f_{i}\in F$ for all i. That is, the set

\{1,\alpha ,\ldots ,{\alpha }^{n-1}\}

is a basis for E as a vector space over F.

Example

If one adjoins to the rational numbers $F=\mathbb {Q}$ the two irrational numbers ${\sqrt {2}}$ and ${\sqrt {3}}$ to get the extension field $E=\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})$ of degree 4, one can show this extension is simple, meaning $E=\mathbb {Q} (\alpha )$ for a single $\alpha \in E$ . Taking $\alpha ={\sqrt {2}}+{\sqrt {3}}$ , the powers 1, α, α², α³ can be expanded as linear combinations of 1, ${\sqrt {2}}$ , ${\sqrt {3}}$ , ${\sqrt {6}}$ with integer coefficients. One can solve this system of linear equations for ${\sqrt {2}}$ and ${\sqrt {3}}$ over $\mathbb {Q} (\alpha )$ , to obtain ${\sqrt {2}}={\tfrac {1}{2}}(\alpha ^{3}-9\alpha )$ and ${\sqrt {3}}=-{\tfrac {1}{2}}(\alpha ^{3}-11\alpha )$ . This shows that α is indeed a primitive element:

\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})=\mathbb {Q} ({\sqrt {2}}+{\sqrt {3}}).

Theorem statement

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

For a non-separable extension $E/F$ 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 $E=\mathbb {F} _{p}(T,U)$ , the field of rational functions in two indeterminates T and U over the finite field with p elements, and $F=\mathbb {F} _{p}(T^{p},U^{p})$ . In fact, for any α = g(T,U) in $E\setminus F$ , the Frobenius endomorphism shows that the element α^p lies in F , so α is a root of $f(X)=X^{p}-\alpha ^{p}\in F[X]$ , and α cannot be a primitive element (of degree p² over F), but instead F(α) is a non-trivial intermediate field.

Proof

Suppose first that $F$ is infinite. If $F(\alpha ,\beta )\supsetneq F(\alpha +c\beta )$ for some $c\in F$ , the latter field must not contain $\beta$ (otherwise $\alpha =(\alpha +c\beta )-c\beta$ would also be in it). Therefore, we may extend the inclusion $F(\alpha +c\beta )\to F(\alpha ,\beta )$ to an $F$ -automorphism $\sigma$ that sends $\beta$ to a different root of the minimal polynomial of $\beta$ over $F(\alpha +c\beta )$ , since they are all different by separability (the polynomial in question is a divisor of the minimal polynomial of $\beta$ over $f$ ). (Possibly we have to enlarge $F(\alpha ,\beta )$ .) We then have

\alpha +c\beta =\sigma (\alpha +c\beta )=\sigma (\alpha )+c\sigma (\beta )

and therefore

c={\frac {\sigma (\alpha )-\alpha }{\beta -\sigma (\beta )}}

.

Since there are only finitely many possibilities for $\sigma$ , there are only finitely many possibilities for the value of $c$ . For all other values of $c\in F$ then $F(\alpha ,\beta )=F(\alpha +c\beta )$ .

For the case where $F$ is finite, we simply use a primitive root, which then generates the field extension.

History

In his First Memoir of 1831, published in 1846,^[1] 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^[2] (as remarked by the referee Siméon Denis Poisson) by exploiting a theorem^[3]^[4] 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.^[4] 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;^[5] Steinitz called the "classical" one 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.^[6]^[7]

References

^ Neumann, Peter M. (2011). The mathematical writings of Évariste Galois. Zürich: European Mathematical Society. ISBN 978-3-03719-104-0. OCLC 757486602.
^ 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.
^ 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.
^ ^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.
^ 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.
^ Kleiner, Israel (2007). "§4.1 Galois theory". A History of Abstract Algebra. Springer. p. 64. ISBN 978-0-8176-4685-1.
^ 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.

External links

[1] Neumann, Peter M. (2011). The mathematical writings of Évariste Galois. Zürich: European Mathematical Society. ISBN 978-3-03719-104-0. OCLC 757486602.

[2] 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.

[3] 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.

[:1-4] Cox, David A. (2012). Galois theory (2nd ed.). Hoboken, NJ: John Wiley & Sons. p. 322. ISBN 978-1-118-21845-7. OCLC 784952441.

[:0-5] 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.

[6] Kleiner, Israel (2007). "§4.1 Galois theory". A History of Abstract Algebra. Springer. p. 64. ISBN 978-0-8176-4685-1.

[7] 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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