for her highly creative work in the theory of
RING
s and
in
ABSTRACT ALGEBRA
. She was responsible for direct-
ing algebra away from detailed arithmetical calcula-
tions to a general axiomatic study of structure. She is
also noted for her determination to succeed as a
woman scholar in the male-dominated realm of early
20th-century academia.
Noether studied languages and music during her
school years, and trained to be a teacher of French and
English. In 1900 she passed her state certification
exams but never worked as an educator. Instead she
decided to pursue a career in mathematics, attending
lectures at the University, of Erlangen. At the time
women were not permitted to enroll as students at the
university, and her studies at the university were never
deemed official. In 1904 she passed the state matricula-
tion exam and then went to the University of Göttin-
gen. Three years later she was awarded a doctorate
from that institution. However, she was not permitted
to pursue the habilitation degree that would earn her
the appropriate qualification to become a university
faculty member. Noether returned to Erlangen.
During the following years Noether published a
number of influential papers on the topic of algebra.
Her work caught the attention of prominent mathe-
maticians D
AVID
H
ILBERT
(1862–1943) and C
HRISTIAN
F
ELIX
K
LEIN
(1849–1925), who invited her back to the
University of Göttingen and fought the university
administration to give her the right to pursue an
advanced degree. She was finally awarded an official
position as a faculty member of the university in 1922.
Mathematicians today consider her 1921 paper
“Idealtheorie in Ringbereichen” (The theory of ideals
in ring structures) to be of fundamental influence in the
development of modern abstract algebra.
Noether was granted many honors and awards for
her work. In 1928 she was invited to address the Inter-
national Mathematical Conference at Bologna, and
spoke again at Zürich in 1932, the same year she was
a joint recipient of the Alfred Ackermann-Teubner
Memorial Prize for the advancement of mathematical
knowledge.
With the uprising of the Nazis in 1933 Noether
was dismissed from her position at Göttingen because
of her Jewish faith. She then accepted a professorship
at Bryn Mawr College, Pennsylvania. At this time she
also lectured at the Institute of Advanced Study at
Princeton, New Jersey.
Noether died at Bryn Mawr on April 14, 1935. She
wrote a total of 45 research papers throughout her
career and shaped the entire course of study in abstract
algebra. Mathematician Bartel Leendert van der Waer-
den (1903–96) was particularly influenced by her work
and continued to promote her ideas after her death.
non-Euclidean geometry After numerous unsuc-
cessful attempts throughout history to establish the
PARALLEL POSTULATE
as a consequence of the remaining
four of E
UCLID
’
S POSTULATES
, mathematicians began to
contemplate theories of geometry in which the fifth
postulate does not hold. Any such theory of
GEOMETRY
is called a non-Euclidean geometry.
In 1795 Scottish mathematician and physicist John
Playfair (1748–1819) presented an alternative, but
equivalent, formulation of the parallel postulate:
through any point in the plane, there is precisely one
line through that point parallel to any prescribed
direction. Recasting the postulate this way makes it
apparent that negation of the famous fifth postulate
has two parts. Either:
1. There are no lines through a given point parallel to
a given direction.
2. There is more than one line through a given point
parallel to a given direction.
From 1826 to 1829 Russian mathematician N
ICO
-
LAI
I
VANOVICH
L
OBACHEVSKY
(1792–1856) developed
a consistent theory of geometry in which Euclid’s fifth
postulate fails in the manner described in point 2.
Hungarian mathematician J
ÁNOS
B
OLYAI
(1802–60)
independently came to the same surprising conclusion
by assuming that through any point there are
infinitely many distinct lines parallel to a given direc-
tion. (C
ARL
F
RIEDRICH
G
AUSS
(1777–18) had also
come to similar conclusions, but he did not publish
his results.) Such a theory of geometry is today called
HYPERBOLIC GEOMETRY
.
In the 1850s German mathematician G
EORG
F
RIEDRICH
B
ERNHARD
R
IEMANN
(1826–66) presented
an alternative form of geometry in which Euclid’s fifth
postulate fails in the manner describe in point 1 above.
In this theory, today called Riemannian geometry or
SPHERICAL GEOMETRY
, “lines” are great circles drawn
on the surface of a sphere. Consequently, it is impossi-
ble to draw a pair of lines that never intersect.
non-Euclidean geometry 355