1911 Encyclopædia Britannica/Number/Normal Fields

51. Normal Fields.—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.