1911 Encyclopædia Britannica/Number/Normal Fields

51. Normal Fields.&mdash;The special properties of a particular field $$\Omega$$ are closely connected with its relations to the conjugate fields $$\Omega', \Omega, \dots \Omega^{(n-1)})$$. The most important case is when each of the conjugate fields is identical with $$\Omega$$: the field is then said to be Galoisian or normal''. The aggregate $$\mathrm{R}(\Omega, \Omega', \dots \Omega^{(n-1)}))$$ of all rational functions of $$\theta$$ and its conjugates is a normal field: hence every arithmetical field of order $$n$$ is either normal, or contained in a normal field of a higher order. The roots of an equation $$f(\theta)=0$$ which defines a normal field are associated with a group of substitutions: if this is Abelian, the field is called Abelian; if it is cyclic, the field is called cyclic. A cyclotomic field is one the elements of which are all expressible as rational functions of roots of unity; in particular the complete cyclotomic field $$\mathrm{C}_m$$, of order $$\phi(m)$$, is the aggregate of all rational functions of a primitive mth root of unity. To Kronecker is due the very remarkable theorem that all Abelian (including cyclic) fields are cyclotomic: the first published proof of this was given by Weber, and another is due to D. Hubert.

Many important theorems concerning a normal field have been established by Hilbert. He shows that if $$\Omega$$ is a given normal field of order $$m$$, and $$\mathfrak{p}$$ any of its prime ideals, there is a finite series of associated fields $$\Omega_1, \Omega_2, \!\And\!\!\!\!\mathrm{c}.$$, of orders $$m_1, m_2, \!\And\!\!\!\!\mathrm{c}.$$, such that $$m_i \equiv 0 \pmod{m_{i+1}}$$, and that if $$r_i = m/m_i$$, $$\mathfrak{p}^{r_i} = \mathfrak{p}_i$$, a prime ideal in $$\Omega_i$$. If $$\Omega_l$$ is the last of this series, it is called the field of inertia (Trägheitskörper) for $$\mathfrak{p}$$: next after this comes another field of still lower order called the resolving field (Zerlegungskörper) for $$\mathfrak{p}$$, and in this field there is a prime of the first degree, $$\mathfrak{p}_{l+1}$$, such that $$\mathfrak{p}_{l+1}=\mathfrak{p}^k$$, where $$k = m/m_l$$. In the field of inertia $$\mathfrak{p}_{l+1}$$ remains a prime, but becomes of higher degree; in $$\Omega_{l-1}$$, which is called the branch-field (Verzweigungskörper) it becomes a power of a prime, and by going on in this way from the resolving field to $$\Omega$$, we obtain $$(l+2)$$ representations for any prime ideal of the resolving field. By means of these theorems, Hilbert finds an expression for the exact power to which a rational prime $$p$$ occurs in the discriminant of $$\Omega$$, and in other ways the structure of $$\Omega$$ becomes more evident. It may be observed that when $$m$$ is prime the whole series reduces to $$\Omega$$ and the rational field, and we conclude that every prime ideal in $$\Omega$$ is of the first or mth degree: this is the case, for instance, when $$m=2$$, and is one of the reasons why quadratic fields are comparatively so simple in character.