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field [2013/12/21 19:57] 127.0.0.1 external edit |
field [2014/09/28 11:51] nikolaj |
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| ===== Field ===== | ===== Field ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $\langle X,+,* \rangle \in \mathrm{field}(X)$ | | |
| - | | @#FFBB00: $\langle X,+,* \rangle \in \mathrm{field}(X)$ | | + | | @#55EE55: postulate | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ | |
| - | + | | @#55EE55: postulate | @#55EE55: $\langle X,* \rangle \in \mathrm{abelianGroup}(X)$ | | |
| - | | @#55EE55: $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ | | + | |
| - | | @#55EE55: $\langle X,* \rangle \in \mathrm{abelianGroup}(X)$ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
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| One generally (also) calls $F$ the field. | One generally (also) calls $F$ the field. | ||
| + | |||
| + | === Theorems === | ||
| + | Finite fields are completely determined by their cardinality. | ||
| + | |||
| + | There is a field of cardinality for each $p^n$, $p$ prime. | ||
| + | |||
| ==== Reference ==== | ==== Reference ==== | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Field_%28mathematics%29|Field]] | Wikipedia: [[http://en.wikipedia.org/wiki/Field_%28mathematics%29|Field]] | ||
