• 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 solidi±ed the ±eld. • Separate British/American and Continental European threads. • Groups ◦ De±nition: 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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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.