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| Preface | |
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| Algebraic preliminaries | |
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| Groups, fields and vector spaces | |
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| Groups | |
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| Fields | |
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| Vector spaces | |
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| The axiom of choice, and Zorn's lemma | |
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| The axiom of choice | |
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| Zorn's lemma | |
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| The existence of a basis | |
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| Rings | |
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| Rings | |
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| Integral domains | |
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| Ideals | |
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| Irreducibles, primes and unique factorization domains | |
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| Principal ideal domains | |
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| Highest common factors | |
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| Polynomials over unique factorization domains | |
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| The existence of maximal proper ideals | |
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| More about fields | |
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| The theory of fields, and Galois theory | |
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| Field extensions | |
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| Introduction | |
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| Field extensions | |
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| Algebraic and transcendental elements | |
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| Algebraic extensions | |
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| Monomorphisms of algebraic extensions | |
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| Tests for irreducibility | |
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| Introduction | |
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| Eisenstein's criterion | |
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| Other methods for establishing irreducibility | |
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| Ruler-and-compass constructions | |
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| Constructible points | |
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| The angle [pi]/3 cannot be trisected | |
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| Concluding remarks | |
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| Splitting fields | |
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| Splitting fields | |
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| The extension of monomorphisms | |
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| Some examples | |
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| The algebraic closure of a field | |
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| Introduction | |
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| The existence of an algebraic closure | |
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| The uniqueness of an algebraic closure | |
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| Conclusions | |
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| Normal extensions | |
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| Basic properties | |
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| Monomorphisms and automorphisms | |
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| Separability | |
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| Basic ideas | |
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| Monomorphisms and automorphisms | |
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| Galois extensions | |
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| Differentiation | |
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| The Frobenius monomorphism | |
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| Inseparable polynomials | |
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| Automorphisms and fixed fields | |
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| Fixed fields and Galois groups | |
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| The Galois group of a polynomial | |
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| An example | |
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| The fundamental theorem of Galois theory | |
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| The theorem on natural irrationalities | |
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| Finite fields | |
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| A description of the finite fields | |
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| An example | |
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| Some abelian group theory | |
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| The multiplicative group of a finite field | |
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| The automorphism group of a finite field | |
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| The theorem of the primitive element | |
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| A criterion in terms of intermediate fields | |
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| The theorem of the primitive element | |
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| An example | |
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| Cubics and quartics | |
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| Extension by radicals | |
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| The discriminant | |
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| Cubic polynomials | |
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| Quartic polynomials | |
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| Roots of unity | |
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| Cyclotomic polynomials | |
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| Irreducibility | |
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| The Galois group of a cyclotomic polynomial | |
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| Cyclic extensions | |
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| A necessary condition | |
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| Abel's theorem | |
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| A sufficient condition | |
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| Kummer extensions | |
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| Solution by radicals | |
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| Soluble groups: examples | |
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| Soluble groups: basic theory | |
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| Polynomials with soluble Galois groups | |
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| Polynomials which are solvable by radicals | |
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| Transcendental elements and algebraic independence | |
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| Transcendental elements and algebraic independence | |
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| Transcendence bases | |
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| Transcendence degree | |
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| The tower law for transcendence degree | |
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| Luroth's theorem | |
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| Some further topics | |
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| Generic polynomials | |
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| The normal basis theorem | |
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| Constructing regular polygons | |
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| The calculation of Galois groups | |
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| A procedure for determining the Galois group of a polynomial | |
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| The soluble transitive subgroups of [Sigma subscript p] | |
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| The Galois group of a quintic | |
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| Concluding remarks | |
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| Index | |