六十年前的考研究试题
2020-03-30 12:36阅读：
六十年前的考研究试题
记得，上世纪60年代初，中科院东湖数学所招收代数喜专业研究生（指导教师华罗庚教授），袁萌应考。
华罗庚教授为代数考卷亲自命题
代数考卷的第一道考试题是：证明区裙{0,1}必定是域。
你晕不晕？
请见附件。
袁萌 陈启清
5月30日
附件：
In mathematics, a group（群） is a set
equipped with a binary operation that combines any two elements to
form a third element in such a way that four conditions called
group axioms are satisfied, namely closure, associativity, identity
and invertibility. One of the most familiar examples of a group is
the set of integers together with the addition operation, but
groups are encountered
in numerous areas within and outside mathematics, and help focusing
on essential structural aspects, by detaching them from the
concrete nature of the subject of the study.[1][2]
Groups share a fundamental kinship with the notion of
symmetry. For example, a symmetry group encodes symmetry features
of a geometrical object: the group consists of the set of
transformations that leave the object unchanged and the operation
of combining two such transformations by performing one after the
other. Lie groups are the symmetry groups used in the Standard
Model of particle physics; Poincaré groups, which are also Lie
groups, can express the physical symmetry underlying special
relativity; and point groups are used to help understand symmetry
phenomena in molecular chemistry.
The concept of a group arose from the study of polynomial
equations, starting with Évariste Galois in the 1830s, who
introduced the term of group (groupe, in French) for the symmetry
group of the roots of an equation, now called a Galois group. After
contributions from other fields such as number theory and geometry,
the group notion was generalized and firmly established around
1870. Modern group theory—an active mathematical discipline—studies
groups in their own right.[a] To explore groups, mathematicians
have devised various notions to break groups into smaller,
better-understandable pieces, such as subgroups, quotient groups
and simple groups. In addition to their abstract properties, group
theorists also study the different ways in which a group can be
expressed concretely, both from a point of view of representation
theory (that is, through the representations of the group) and of
computational group theory. A theory has been developed for finite
groups, which culminated with the classification of finite simple
groups, completed in 2004.[aa] Since the mid-1980s, geometric group
theory, which studies finitely generated groups as geometric
objects, has become an active area in group theory.
Algebraic structure → Group
theory
Group theory
Basic notions
[show]
