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.
Let
E/F
\alpha\inE
E/F
E=F(\alpha),
E
\alpha
F
E/F
If the field extension
E/F
\alpha
n=[E:F]
\gamma\inE
\gamma=a0+a1{\alpha}+ … +an-1{\alpha}n-1,
for unique coefficients
a0,a1,\ldots,an-1\inF
\{1,\alpha,\ldots,{\alpha}n-1\}
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
f(X)\inF[X]
\{1,\alpha,\ldots,\alphan-1,\alphan\}
If L is a splitting field of
f(X)
\alpha1,\ldots,\alphan
\sigmai:F(\alpha)\hookrightarrowL
\sigmai(\alpha)=\alphai
\sigma(a)=a
a\inF
\sigma1,\ldots,\sigman\inGal(L/F)
[E:F]=n
\alpha
\alpha
\sigma1(\alpha),\ldots,\sigman(\alpha)
L\supseteqE
If one adjoins to the rational numbers
F=Q
\sqrt{2}
\sqrt{3}
E=Q(\sqrt{2},\sqrt{3})
E=Q(\alpha)
\alpha\inE
\alpha=\sqrt{2}+\sqrt{3}
\sqrt{2}
\sqrt{3}
\sqrt{6}
\sqrt{2}
\sqrt{3}
Q(\alpha)
\sqrt{2}=\tfrac12(\alpha3-9\alpha)
\sqrt{3}=-\tfrac12(\alpha3-11\alpha)
Q(\sqrt2,\sqrt3)=Q(\sqrt2+\sqrt3).
E=\Q(\sqrt2,\sqrt3)
\sigma1,\sigma2,\sigma3,\sigma4:E\toE
\sigmai(\sqrt2)=\pm\sqrt2
\sigmai(\sqrt3)=\pm\sqrt3
f(X)\in\Q[X]
\alpha=\sqrt2+\sqrt3
f(\sigmai(\alpha))=\sigmai(f(\alpha))=0
f(X)
\sigmai(\alpha)=\pm\sqrt2\pm\sqrt3
f(X)
[\Q(\alpha):\Q]\geq4
[E:\Q]=4
E=\Q(\alpha)
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.
For a non-separable extension
E/F
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=Fp(T,U)
p,U | |
F=F | |
p(T |
p)
\alpha=g(T,U)
E\setminusF
\alphap
f(X)=Xp-\alphap\inF[X]
Suppose first that
F
E=F(\beta,\gamma)
c\inF
\alpha=\beta+c\gamma
F(\alpha)\subsetneqF(\beta,\gamma)
\gamma\notinF(\alpha)
\beta=\alpha-c\gamma\inF(\alpha)=F(\beta,\gamma)
\beta,\gamma
F(\alpha)
f(X),g(X)\inF(\alpha)[X]
L
\beta,\beta',\ldots
f(X)
\gamma,\gamma',\ldots
g(X)
\gamma\notinF(\alpha)
\gamma' ≠ \gamma
\sigma:L\toL
F(\alpha)
\sigma(\gamma)=\gamma'
\sigma(\alpha)=\alpha
\beta+c\gamma=\sigma(\beta+c\gamma)=\sigma(\beta)+c\sigma(\gamma)
c=
\sigma(\beta)-\beta | |
\gamma-\sigma(\gamma) |
\sigma(\beta)=\beta'
\sigma(\gamma)=\gamma'
c\inF
\alpha=\beta+c\gamma
F(\alpha)=F(\beta,\gamma)
For the case where
F
\alpha
E
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. 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]