Appendix A
Published online by Cambridge University Press: 05 June 2012
Summary
Fields
Fields are important algebraic structures used in almost all branches of mathematics. Here we only cover the definitions and theorems needed for the purposes of this book.
Definition. A field F is a set along with two operations (denoted with addition and multiplication notation) on pairs of elements of F such that the following properties are satisfied.
For all a and b in F, we have that a + b ∈ F.
For all a, b, and c in F, we have that (a + b) + c = a + (b + c).
There exists an element 0 in F satisfying a + 0 = a for all a ∈ F.
For every a ∈ F there exists a b in F such that a + b = 0.
For all a and b in F we have that a + b = b + a.
For all a and b in F we have that ab ∈ F.
For all a, b, and c in F we have that (ab)c = a(bc).
There is an element 1 in F satisfying 1a = a for all a ∈ F.
For every a ∈ F with a ≠ = 0, there exists a b ∈ F such that ab = 1.
For every a and b in F we have that ab = ba.
For every a, b, and c in F we have that a(b + c) = ab + ac.
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- Protecting InformationFrom Classical Error Correction to Quantum Cryptography, pp. 269 - 276Publisher: Cambridge University PressPrint publication year: 2006