•
In 1830, George Peacock tried to systematize the study of algebra (like Euclid did for ge
ometry).
He had a Principle of Permanence of Form that was analogous to the geometric
Principle of Continuity. Boyer: “The beginnings of postulational thinking in arithmetic and
algebra”.
•
Abel and Galois laid the groundwork for these abstract structures but didn’t explicitly dis
cuss the structures themselves.
•
Van der Waerden’s
Modern Algebra
in 1930 really solidified the field.
•
Separate British/American and Continental European threads.
•
Groups
◦
Definition: A group is a set
G
with a binary operation
∗
:
G
×
G
→
G
such that:
⋆
∗
is associative:
a
∗
(b
∗
c)
=
(a
∗
b)
∗
c
for every
a, b, c
∈
G
.
⋆
there is an identity
e
∈
G
such that
e
∗
g
=
g
∗
e
=
e
for every
g
∈
G
.
⋆
Every
g
∈
G
has an inverse
g
−
1
∈
G
with the property that
g
∗
g
−
1
=
g
−
1
∗
g
=
e
.
◦
Motivation: Solvability of Equations, Classical Geometric Constructions problems.
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 Fall '10
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 Algebra, Geometry, Continuity, Classical Geometric Constructions, Geometric Constructions problems

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