Steinitz, E.

Japanese: Steinitz，E.（英語表記）SteinitzE

...In the latter half of the 18th century, the properties of roots were elucidated along with research into solving higher-order equations, and in the 1820s, E. Galois came up with the idea of "fields with roots of equations" and what would be called today's "Galois groups of equations," and showed the correspondence between subgroups and subfields. The results were generalized to the case of finite algebraic extension fields by JW Dedekind towards the end of the 19th century, and in 1910 by Ernst Steinitz (1871-1928), and in 1928, Wolfgang Krull (1899-1976) generalized it to the case of infinite algebraic extensions using the concept of topological groups. There are applications of these results, as well as generalizations to the Galois theory of rings, but we will focus on the most basic case of number fields. ...

From [Body]

…L. Kronecker introduced the idea of giving a finite algebraic extension field of a field K in the form of a coset ring K [ x ] / f ( x ) K [ x ] by an irreducible polynomial f ( x ) of a polynomial ring K[x]. In addition, the congruence modulo p , which has long been used to treat integers, has made it possible to consider fields consisting of p (prime) elements. Furthermore, with the introduction of p -adic numbers by K. Hensel (1861-1941), E. Steinitz (1871-1928) unified the theory of fields, including the introduction of concepts such as prime fields, separable algebraic elements, and perfect fields. After that, theories such as infinite algebraic extension and transcendental extension were further developed. …

*Some of the terminology references "Steinitz, E." are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…高次方程式の解法に関する研究に伴って，18世紀後半に根の性質の解明が進み，1820年代に入ってから，É.ガロアが〈方程式の根をつけ加えた体〉および今日の言葉で〈方程式のガロア群〉と呼ばれるものを考え，部分群と部分体との対応を示した。その結果は，19世紀末ごろJ.W.デデキント，1910年シュタイニッツErnst Steinitz(1871‐1928)によって有限次代数拡大体の場合に一般化され，また，28年にはクルルWolfgang Krull(1899‐1976)が位相群の概念を利用して，無限次代数拡大の場合に一般化した。　これらの結果の応用，あるいは環のガロア理論への一般化などがあるが，最も基本的な数体の場合を中心にして説明する。…

【体】より

…L.クロネッカーは体Kの有限次代数拡大体を，多項式環K[x]の既約多項式f(x)によって，剰余類環K[x]/f(x)K[x]の形で与える考えを導入した。また古くから整数の扱いにあった〈pを法とする合同〉によって，p(素数)個の元からなる体も考察の対象になり，またヘンゼルK.Hensel(1861‐1941)のp進数の登場などにより，シュタイニッツE.Steinitz(1871‐1928)が，素体，分離代数的元，完全体などの概念の導入を含めて，体の理論を一つのまとまった形にした。その後，無限次の代数拡大や超越拡大などの理論がさらに整備された。…