Emmy Noether

“Of
all the women mathematicians, Emmy Noether is generally the best known. Often
described as a loving, intelligent woman, she was impressive by many standards.
She was faced with gender issues and political tensions in her lifetime, but
her passion for mathematics remained strong.”(Mishna par. 1)

Emmy Noether’s early years got her started in
her interest of learning. Emmy Noether was born on March 23, 1882 in Erlangen
Germany to Max and Ida Noether. She was raised as a typical middle class German
daughter and went to school from age seven to fifteen. She pursued further
study in French and English. By the age of eighteen she passed the examination
of the state of Bavaria for teachers of English and French at schools for girls.
Emmy was not satisfied ending her education so she decided to try and continue
her education.

Emmy
had to struggle to get the higher education that she wanted. Starting in 1900
women were allowed to enroll in universities only as auditors. Following in her
father’s footsteps she decided to study mathematics. From 1900 to 1902 Emmy was
an auditor for mathematics classes at Erlangen while she studied for the
Absolutorium (high school certification). One major problem for women during
this time was receiving the proper education to be able to enter college. In
1903 Emmy got her certification to enroll as an auditor at Gottingen and was
able to attend lectured by Otto Blumenthal, David Hilbert, Felix Klein, and
Hermann Minkowski. She soon returned to Erlangen so she would be able to be a
regular student. In December of 1907 Emmy earned her Ph.D. in mathematics. She
wrote her thesis under Paul Gordan on complete systems of invariants of ternary
biquadric forms. Gordan inspired Emmy in studies of mathematics. Gordon was a
strong believer in being straightforward in the writings of mathematics. In his
papers a reader could go pages before reading any text.

For the next eight years Emmy researched and
substituted for her dad at Erlangen. Emmy did not have any opportunities after
she received her doctorate. Women were not allowed to teach in universities so
Emmy could not make a living. Her dad allowed her to substitute under his name
but she still could not make a salary.

After
working with her father she started to become well known and working with
renowned mathematicians. In 1909 Emmy joined the German Mathematical
Association and gave her first public talk establishing herself as a
mathematician. From 1910 to 1919 Ernst Fischer had a greater influence then anyone
on Emmy’s mathematics. Together they studied finite rational and integral
bases. In 1916 she derived the conservation laws of physics that made her well
known in the physics realm. Hilbert and Klein invited Emmy back to Gottingen in
1915. Emmy’s attempt to obtain Habilitation (permission to lecture) in 1915 was
denied. Hilbert allowed her to lecture under his name until 1919 when women
were granted habilitation. Unfortunately for Emmy she was not given a salary
for the work that she did. In 1922 she was recommended for associate professor
without tenure. With this promotion she received a small salary. From 1924-1925
B.L. van der Waerden became one of Emmy’s students.

As Emmy’s career continued she obtained students
who loved her studies and managed to help her as well. From 1924-1925 B.L. van
der Waerden became one of Emmy’s students. He was the popularizer of her work.
Emmy’s lectures were meant for small groups because she was not a very good
public speaker. Van der Waerden made things clearer so individuals could
understand her work better. In 1927 Max Deuring became her pupil during her
work of non-commutative algebra and hypercomplex areas.

Emmy’s
career started ending once Germany started having its political struggles. In
1932, after the Nazis took over, she lost her job. In addition to losing her
job, the Gottingen School of Algebra, which had been founded by Emmy, was
destroyed. She then moved to the United States and in August of 1933 where she
started giving lectures at Bryn Mawr and weekly lectures at the Institute for
the Advanced Study in Princeton. During her second year she suddenly died after
and operation for removal of a tumor.

Emmy
Noether was an astounding mathematician. In 1932 Emmy was awarded the Alfred
Ackermann-Teubner Memorial Prize for the Advancement of the Mathematical
Sciences. She worked hard to find simpler ways of understanding algebra.
Unfortunately, Emmy’s lectures were not made for large groups. Her delivery of
her work was poor, hurried, and inconsistent.

Emmy Noether overcame a lot of prejudice to
become the well-known mathematician that she was. As a woman she was held back
in her studies as well as her right to lecture. She was very fortunate to have
friends like Hilbert that helped her overcome these obstacles. She was also
passed up for the Gottingen Academy of Sciences, which several of her
colleagues were elected for. Also as a Jew she faced many prejudices. She was
kicked out of Gottingen and forced to move to the United States because of her
religious beliefs.

“Emmy
Noether was an amazing mathematician. She taught us how to think in simple and
thus general terms.” “She therefore opened a path to the discovery of algebraic
regularities where before the regularities had been obscured by complicated
specific conditions.”(Kimberling, 143). Her greatest talent was toward general
mathematical conceptions.

The mathematics of Emmy Noether

The
aspect of Noethers work that we will be discussing is Noetherian Rings on
Ideals. Her work on Ideals ,in
general, began around 1919-1920.
This is the main work by which many mathematicians know of Noether. But before we can begin talking
about Rings and Ideals we must first learn some terminology that will guide us
through this material. In order to guide you through the several steps we will
take an easy example that can be understood by all levels of mathematicians. We
are going to take you through the steps of learning about a Noetherian ring by
using the integers as our example. There are many examples that we can use such
as the matrices but the integers is the easiest to explain.

•
First
you must find a set. A set is a collection of objects. The objects are either in the set or are
not. Example: {Z} is the set of all integers

•
After
picking the set we must establish if this set is a group. A group is a structure consisting of a
set G and any binary operation on
G. We will use + as our
binary operation and G= {Z} ,all integers, as our set. For + to be our binary operation, Z
must be closed under addition. For
all a,b in Z, a+b must be in Z.
Once we have picked our set and the binary operation we must check the
following three conditions and see if they hold in order to determine if the
set and binary operation make a group.

•
If
a,b,c are elements of G, then a+(b+c)=(a+b)+c which is the Associative
Property. An example of this property
using 2,3,4 which are elements in
{Z} is 2+(3+4)=(2+3)+4 which is true for all {Z} . This is just one
example but you can infer from this that this property will hold for all Z.

•
There
is an element of G, called e, such that for each a element of G, e+a=a; e is a left neutral element for G. This is the Identity Property. An example of this property is e=0 for {Z} , 1+0=1, 2+0=2. This is
just one example but you can see how you can use any integer and this would
hold true.

•
For
each a element of G there is an element b in G such that b+a=e; b is a left inverse of a with respect
to e. This is called the Additive Inverse Property. An example of this property
is as follows. 2 and -2 are in
{Z} -2+2=0=e. This is just one
example but notice it would be true for any integer you used.

You have now given the insight to show why the
integers are a group. Now the next step is to show that this group is abelian.

•
The
next step is to show that the group integers is abelian. For each a and b in G
a+b=b+a for it to be an Abelian group.
This is the Commutative Property.
An example of this property is as follows 2,3 are in {Z} , 2+3=3+2. This is just one example but
notice it would be true for any integer you put into this property.

From this information we have shown that
{Z} is an Abelian Group under +.
Now we must show that this abelian group is a ring.

•
The
next step is to identify if an abelian group is a ring. A ring is a set R on
which are defined by two binary operations + and *
(in that order). In order to use
multiplication as our binary operation, Z must also be closed under
multiplication. For all a,b in Z,
(a)(b) must be in Z. With
these two operations we need to show that the following three properties hold
in order to prove that an abelian group is a ring. In our example we will still
be using {Z} as our set.

•
R
is an Abelian group under +. The netural element of this group is 0. From the information above we have
shown {Z} to follow this rule.

•
Multiplication
is associative. Example: 2,3,4 are in {Z} , 2(3*4)=(2*3)4. This
is just one example but notice it would be true for any element Z.

•
Multiplication
is distributive over addition; for all a,b,c in R a(b+c)=ab+ac
and (b+c)a=ba+ca. Example: 2,3,4 are in {Z} , 2(3+4)=(2)(3)+(2)(4)
and (3+4)2=(3)(2)+(4)(2). This is
just one example but if you put in any integer you will notice that it holds
true.

From the information above we have given the
intuition that {Z} is a Ring. Now
we need to explain what a subset and an Ideal are.

•
A
is a proper subset of B if and
only if all of set A is in set B and A does not equal B. Example: {2Z} , all even integers, is a subset of {Z} because all of {2Z} is in {Z} .

•
A
nonempty subset I of a ring R is an ideal in R if and only if : (use {2Z} as Ideal.) The following two conditions must hold true in
order for the ring to be an ideal.

•
For
all a,b in I, a-b is in I. This is equivalent to the fact that I is a subring
of R. For an example to see if 2Z
is a subring of R we could go back through all the steps in the definition of a
ring , but it is a lot easier to show that above condition. In this case if you
take two integers and subtract them you will still get an even integer. (4
-2=2)

•
For
each y in R and each a in I, ya is in I.
For instance with I=2Z and R=Z 2*3=6, where 6 is even and an element of
Z. A counter example would be if I=2Z+1 and R=Z then 3*2=6 but 6 is not in I
because I includes only the odd integers.

Now that we have found out what an ideal is in a
ring we need to look to see what an ACC of Ideals is.

•
The
ACC of Ideals says that there are infinitely many ideals that get bigger but
they are a descending chain of ideals. In order for the ACC of Ideals to be
noetherian it must have a stopping point For example with the even integers it
stops at 2Z and 2Z contains 4Z which contains 8Z and so forth.

•
A
Notherian Ring is a ring where there is no infinite increasing chain of Ideals
in R. We will give you and idea of why it is also a Noetheian Ring. Similar
ideas to the properties of ideals we will show that the ideals are even
multiples of {2Z} . Now we will give you an idea why other subster of 2Zare not
ideals. Now take {2Z+1} ,or the
odd integers, as our subset. From
the definitions above we can see that {2Z+1} is a subset of {Z} .
But for it to be an Ideal we must be able to take any other number in
{Z} and be able to multiply it by
an element in {2Z+1} , and the answer be a number in {2Z+1} . We can easily show this does not work.
An example is that 3 is an element
of {2Z+1} and 2 is an element of
{Z} . For {2Z+1} to be an Ideal (2)(3)=an odd
integer. But we all know that
(2)(3)=6 which is not an odd integer.
Therefore {2Z+1} is not an
Ideal and can't be used in a Notherian Ring because it violates the Ascending
Chain Condition. {2Z} is the
largest Ideal in {Z} . {2Z} is an Ideal that we showed above. Now take {4Z} . it is contained in {Z} and in {2Z} . Just like this, {2Z}
contains {4Z} contains
{8Z} and etc. But as you can see the Ideals contain
even integers but the number of integers in each Ideal keeps getting smaller
than the one before it. So the
largest Ideal we can get is {2Z} .
This shows that there is a finite stopping place for the Ideals. Which means that {Z} is a Notherian Ring. (All definitions courtesy Landin and
also Dummit).

Now,
we will briefly show a ring which is not Noetherian. This example is based on all continuous functions on the
closed interval from 0 - 1. Lets
assume we have a set defined as follows f:[0,1] to the Reals with
(f+g)(x)=f(x)+g(x) and (f*g)(x)=f(x)*g(x). From the information provided above we can show that f:[0,1]
to the reals is a ring. We can
then also show that this ring is not Noetherian. This is true because it violates the Ascending Chain
Condition. In other words, In={f
in F such that f (1/n) = 0 for all n greater than or equal to N} . This shows that f is an element of I
n+1 such that f is not an element of I n.
So all this just says that as the Ideals of the ring get larger, they never
have a stopping place. This is the
easiest way to show that a ring is not Noetherian.

Now
we will discuss the importance of Emmy Noethers’ work and how it pertains to
her, mathematics and the world around us.
The work that she did with Ideals is one of her biggest claims to fame
as far as mathematicians are concerned.
This phrase from “In Memory
of Emmy Noether” gives us a nice example of the way that Noether related her
math to the people and world around her and to the improtance that her work
had. “I believe that, of
everything that Noether did, it is the foundations of general Ideal theory and
everything connected with this that has had, is continuing to have, and will
have in the future the greatest impact on mathematics as a whole”(158). From the work she did, she opened up
whole new areas of math for future generations to flourish in. “Noetherian Rings are one of the basic
structures of all of mathematics”(Ratliff,169). The definition of a Noetherian Rings can be modified to form
the definition of a Noetherian Modules which are used in vector spaces. With all of us being math majors, we
know how important vector spaces can be.
Noetherian Rings also have Physics applications. Noether made great
strides to show that simple was better.
“She thought that the easiest way to find an answer to a complicated
algebraic guestion was to use simple and general algebraic concepts instead of
cumbersome computations to do so.”(158)
We think that this gives justice to the claim that Emmy Noether was the
Mother of Algebra.

Through
discrimination Emmy Noether fought to become one of the greatest mathematicians
of all time. With her work in
Algebra, she opened the door for new explorations in both math and the
sciences. Without this woman,
todays world would be a different place and one without many of the advantages
that this womans’ work brought to us.
She fought a long battle to become a great mathematician, she fought a
long battle and won.