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Introduction to algebra
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III · Main algebraic structures
Chapter III.5
Beginning of Galois theory
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On this page
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III.5.1 · Finite field extensions
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III.5.2 · Finite fields
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III.5.3 · Galois correspondence
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III.5.4 · Computing the Galois group
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III.5.5 · Galois extensions and related issues
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III.5.6 · Rigidity and rationality in finite groups
III.5.7 · Epilogue
Section III.5.1
Finite field extensions
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Subsection III.5.1.1
Primitive elements and degrees of extensions
Subsection III.5.1.2
Field isomorphism of splitting fields
Subsection III.5.1.3
Existence of a primitive element
Section III.5.2
Finite fields
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Subsection III.5.2.1
Existence and uniqueness
Subsection III.5.2.2
Subfields and automorphisms of a finite field
Subsection III.5.2.3
Möbius inversion formula and its applications
Section III.5.3
Galois correspondence
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Subsection III.5.3.1
Preliminary results
Subsection III.5.3.2
Fundamental Galois correspondence
Subsection III.5.3.3
Illustrations for the Galois correspondence
Section III.5.4
Computing the Galois group
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Subsection III.5.4.1
The action of the group
Gal
(
f
)
\operatorname{Gal}(f)
Gal
(
f
)
on the roots of the polynomial
f
f
f
Subsection III.5.4.2
Polynomials and groups of prime degree
Subsection III.5.4.3
Reduction method modulo
p
p
p
Subsection III.5.4.4
Normal basis
Section III.5.5
Galois extensions and related issues
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Subsection III.5.5.1
Prime numbers in an arithmetic progression
Subsection III.5.5.2
Galois extensions with an abelian Galois group
Subsection III.5.5.3
Norm and trace
Subsection III.5.5.4
Cyclic extensions
Subsection III.5.5.5
The criterion for solvability of equations in radicals
Section III.5.6
Rigidity and rationality in finite groups
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Subsection III.5.6.1
Definitions and statement of the main theorem
Subsection III.5.6.2
Counting solutions
Subsection III.5.6.3
Examples of rigidity
Section III.5.7
Epilogue