Degree of a field extension From Wikipedia, the free encyclopedia Contents 1 Abstract algebra 1 1.1 History ................................................. 1 1.1.1 Early group theory ...................................... 1 1.1.2 Modern algebra ........................................ 3 1.2 Basic concepts ............................................. 4 1.3 Applications .............................................. 4 1.4 See also ................................................ 5 1.5 References ............................................... 5 1.6 Sources ................................................ 5 1.7 External links ............................................. 5 2 Adjunction (field theory) 6 2.1 Definition ............................................... 6 2.2 Notes ................................................. 6 2.3 Examples ............................................... 6 2.4 Properties ............................................... 6 2.5 References ............................................... 7 3 Algebraic element 8 3.1 Examples ............................................... 8 3.2 Properties ............................................... 8 3.3 See also ................................................ 8 3.4 References ............................................... 9 4 Algebraic extension 10 4.1 Properties ............................................... 10 4.2 Generalizations ............................................ 11 4.3 See also ................................................ 11 4.4 Notes ................................................. 11 4.5 References ............................................... 11 5 Algebraic structure 12 5.1 Introduction .............................................. 12 5.2 Examples ............................................... 12 i ii CONTENTS 5.2.1 One set with operations .................................... 12 5.2.2 Two sets with operations ................................... 14 5.3 Hybrid structures ........................................... 15 5.4 Universal algebra ........................................... 15 5.5 Category theory ............................................ 16 5.6 See also ................................................ 16 5.7 References ............................................... 16 5.8 External links ............................................. 17 6 Arity 18 6.1 Examples ............................................... 18 6.1.1 Nullary ............................................ 18 6.1.2 Unary ............................................. 19 6.1.3 Binary ............................................. 19 6.1.4 Ternary ............................................ 19 6.1.5 n-ary ............................................. 19 6.1.6 Variable arity ......................................... 19 6.2 Other names .............................................. 19 6.3 See also ................................................ 20 6.4 References ............................................... 20 6.5 External links ............................................. 21 7 Closure (mathematics) 22 7.1 Basic properties ............................................ 22 7.2 Closed sets ............................................... 22 7.3 P closures of binary relations ..................................... 23 7.4 Closure operator ............................................ 23 7.5 Examples ............................................... 24 7.6 See also ................................................ 24 7.7 Notes ................................................. 24 7.8 References ............................................... 25 8 Degree of a field extension 26 8.1 Definition and notation ........................................ 26 8.2 The multiplicativity formula for degrees ............................... 26 8.2.1 Proof of the multiplicativity formula in the finite case .................... 27 8.2.2 Proof of the formula in the infinite case ........................... 27 8.3 Examples ............................................... 28 8.4 Generalization ............................................ 28 8.5 References .............................................. 28 9 Field (mathematics) 29 9.1 Definition and illustration ....................................... 29 CONTENTS iii 9.1.1 First example: rational numbers ............................... 30 9.1.2 Second example: a field with four elements .......................... 31 9.1.3 Alternative axiomatizations .................................. 31 9.2 Related algebraic structures ...................................... 31 9.2.1 Remarks ........................................... 31 9.3 History ................................................. 32 9.4 Examples ............................................... 32 9.4.1 Rationals and algebraic numbers ............................... 32 9.4.2 Reals, complex numbers, and p-adic numbers ........................ 32 9.4.3 Constructible numbers .................................... 33 9.4.4 Finite fields .......................................... 33 9.4.5 Archimedean fields ...................................... 34 9.4.6 Field of functions ....................................... 34 9.4.7 Local and global fields .................................... 34 9.5 Some first theorems .......................................... 34 9.6 Constructing fields ........................................... 35 9.6.1 Closure operations ...................................... 35 9.6.2 Subfields and field extensions ................................. 35 9.6.3 Rings vs fields ......................................... 36 9.6.4 Ultraproducts ......................................... 36 9.7 Galois theory ............................................. 36 9.8 Generalizations ............................................ 37 9.8.1 Exponentiation ........................................ 37 9.9 Applications .............................................. 37 9.10 See also ................................................ 37 9.11 Notes ................................................. 38 9.12 References ............................................... 38 9.13 Sources ................................................ 38 9.14 External links ............................................. 39 10 Field extension 40 10.1 Definitions .............................................. 40 10.2 Caveats ................................................ 40 10.3 Examples ............................................... 41 10.4 Elementary properties ......................................... 41 10.5 Algebraic and transcendental elements and extensions ........................ 41 10.6 Normal, separable and Galois extensions ............................... 42 10.7 Generalizations ............................................ 43 10.8 Extension of scalars .......................................... 43 10.9 See also ................................................ 43 10.10Notes ................................................. 43 10.11References ............................................... 43 iv CONTENTS 10.12External links ............................................. 43 11 Function (mathematics) 44 11.1 Introduction and examples ....................................... 46 11.2 Definition ............................................... 47 11.3 Notation ................................................ 48 11.4 Specifying a function ......................................... 49 11.4.1 Graph ............................................. 49 11.4.2 Formulas and algorithms ................................... 49 11.4.3 Computability ......................................... 50 11.5 Basic properties ............................................ 50 11.5.1 Image and preimage ...................................... 50 11.5.2 Injective and surjective functions ............................... 51 11.5.3 Function composition ..................................... 51 11.5.4 Identity function ....................................... 53 11.5.5 Restrictions and extensions .................................. 53 11.5.6 Inverse function ........................................ 53 11.6 Types of functions ........................................... 53 11.6.1 Real-valued functions ..................................... 54 11.6.2 Further types of functions ................................... 54 11.7 Function spaces ............................................ 54 11.7.1 Currying ........................................... 55 11.8 Variants and generalizations ...................................... 55 11.8.1 Alternative definition of a function .............................. 55 11.8.2 Partial and multi-valued functions .............................. 55 11.8.3 Functions with multiple inputs and outputs .......................... 56 11.8.4 Functors ............................................ 57 11.9 History ................................................. 57 11.10See also ................................................ 57 11.11Notes ................................................. 57 11.12References ............................................... 58 11.13Further reading ............................................ 58 11.14External links ............................................. 59 12 Galois theory 60 12.1 Application to classical problems ................................... 60 12.2 History ................................................ 60 12.2.1 Pre-history .......................................... 60 12.2.2 Galois’ writings ........................................ 62 12.2.3 Aftermath .........................................