Milne's notes have exercises at the end of every chapter, a chapter of review exercises, and a two-hour exam; solutions (or at least hints) for all of these are given at the end. A lot of the action takes place over $\mathbf Q$, but I saw a fair number of questions about finite fields and they seemed good.
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Before the nineteenth century, algebra was defined as the study of polynomials.[2] Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of algebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal axiomatic definitions of various algebraic structures such as groups, rings, and fields.[3] This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's Moderne Algebra,[4] which start each chapter with a formal definition of a structure and then follow it with concrete examples.[5]
Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic led to the Galois group of a polynomial. Gauss's 1801 study of Fermat's little theorem led to the ring of integers modulo n, the multiplicative group of integers modulo n, and the more general concepts of cyclic groups and abelian groups. Klein's 1872 Erlangen program studied geometry and led to symmetry groups such as the Euclidean group and the group of projective transformations. In 1874 Lie introduced the theory of Lie groups, aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group of Möbius transformations, and its subgroups such as the modular group and Fuchsian group, based on work on automorphic functions in analysis.[10]
For commutative rings, several areas together led to commutative ring theory.[26] In two papers in 1828 and 1832, Gauss formulated the Gaussian integers and showed that they form a unique factorization domain (UFD) and proved the biquadratic reciprocity law. Jacobi and Eisenstein at around the same time proved a cubic reciprocity law for the Eisenstein integers.[25] The study of Fermat's last theorem led to the algebraic integers. In 1847, Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the cyclotomic fields were UFDs, yet as Kummer pointed out, Q ( ζ 23 ) ) \displaystyle \mathbb Q (\zeta _23)) was not a UFD.[27] In 1846 and 1847 Kummer introduced ideal numbers and proved unique factorization into ideal primes for cyclotomic fields.[28] Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of prime ideals, a precursor of the theory of Dedekind domains. Overall, Dedekind's work created the subject of algebraic number theory.[29]
In 1801 Gauss introduced binary quadratic forms over the integers and defined their equivalence. He further defined the discriminant of these forms, which is an invariant of a binary form. Between the 1860s and 1890s invariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the Jacobian and the Hessian for binary quartic forms and cubic forms.[33] In 1868 Gordan proved that the graded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis.[34] Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to Hilbert's basis theorem.[35]
In 1801 Gauss introduced the integers mod p, where p is a prime number. Galois extended this in 1830 to finite fields with p n \displaystyle p^n elements.[43] In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations,[44] the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893.[45] In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms. [46] The first clear definition of an abstract field was due to Heinrich Martin Weber in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry.[47] In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by their characteristic, and proved many theorems commonly seen today.[48]
Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem. 2ff7e9595c
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