Geometry of the Quintic
by Jerry Shurman
Publisher: Wiley-Interscience 1997
Number of pages: 208
The text demonstrates the use of general concepts by applying theorems from various areas in the context of one problem -- solving the quintic. This book helps students at the advanced undergraduate and beginning graduate levels to develop connections between the algebra, geometry, and analysis that they know, and to better appreciate the totality of what they have learned.
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by Mark Reeder - Boston College
From the table of contents: Basic ring theory, polynomial rings; Finite fields; Extensions of rings and fields; Computing Galois groups of polynomials; Galois groups and prime ideals; Cyclotomic extensions and abelian numbers.
by K.G. Ramanathan - Tata Institute of Fundamental Research
These lecture notes on Field theory are aimed at providing the beginner with an introduction to algebraic extensions, algebraic function fields, formally real fields and valuated fields. We assume a familiarity with group theory and vector spaces.
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A concise treatment of Galois theory and the theory of fields, including transcendence degrees and infinite Galois extensions. Contents: Basic definitions and results; Splitting fields; The fundamental theorem of Galois theory; etc.
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The main result is the Hasse principle for the one-dimensional Galois cohomology of simply connected classical groups over number fields. For most groups, this result is closely related to other types of Hasse principle.