Difference between revisions of "09-240/Classnotes for Tuesday September 15"
From Drorbn
(→Examples: note for "iff") |
(→Examples: Addition and multiplication tables.) |
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# <math>\,\!F_7 = \{ 0, 1,2,3,4,5,6 \}</math> | # <math>\,\!F_7 = \{ 0, 1,2,3,4,5,6 \}</math> | ||
# <math>\,\!F_6 = \{ 0, 1,2,3,4,5 \}</math> is not a field (counterexample) | # <math>\,\!F_6 = \{ 0, 1,2,3,4,5 \}</math> is not a field (counterexample) | ||
+ | |||
+ | {| | ||
+ | | | ||
+ | {| border="1" cellspacing="0" | ||
+ | |+ Ex. 4 | ||
+ | |- | ||
+ | ! + !! 0 !! 1 | ||
+ | |- | ||
+ | ! 0 || 0 || 1 | ||
+ | |- | ||
+ | ! 1 || 1 || 0 | ||
+ | |- | ||
+ | |} | ||
+ | |||
+ | | | ||
+ | {| border="1" cellspacing="0" | ||
+ | |+ Ex. 4 | ||
+ | |- | ||
+ | ! × !! 0 !! 1 | ||
+ | |- | ||
+ | ! 0 || 0 || 0 | ||
+ | |- | ||
+ | ! 1 || 0 || 1 | ||
+ | |} | ||
+ | |||
+ | |- | ||
+ | | | ||
+ | {| border="1" cellspacing="0" | ||
+ | |+ Ex. 5 | ||
+ | |- | ||
+ | ! + !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 | ||
+ | |- | ||
+ | ! 0 || 0 || 1 || 2 || 3 || 4 || 5 || 6 | ||
+ | |- | ||
+ | ! 1 || 1 || 2 || 3 || 4 || 5 || 6 || 0 | ||
+ | |- | ||
+ | ! 2 || 2 || 3 || 4 || 5 || 6 || 0 || 1 | ||
+ | |- | ||
+ | ! 3 || 3 || 4 || 5 || 6 || 0 || 1 || 2 | ||
+ | |- | ||
+ | ! 4 || 4 || 5 || 6 || 0 || 1 || 2 || 3 | ||
+ | |- | ||
+ | ! 5 || 5 || 6 || 0 || 1 || 2 || 3 || 4 | ||
+ | |- | ||
+ | ! 6 || 6 || 0 || 1 || 2 || 3 || 4 || 5 | ||
+ | |} | ||
+ | |||
+ | | | ||
+ | {|border="1" cellspacing="0" | ||
+ | |+ Ex. 5 | ||
+ | |- | ||
+ | ! × !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 | ||
+ | |- | ||
+ | ! 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 | ||
+ | |- | ||
+ | ! 1 || 0 || 1 || 2 || 3 || 4 || 5 || 0 | ||
+ | |- | ||
+ | ! 2 || 0 || 2 || 4 || 6 || 1 || 3 || 1 | ||
+ | |- | ||
+ | ! 3 || 0 || 3 || 6 || 2 || 5 || 1 || 2 | ||
+ | |- | ||
+ | ! 4 || 0 || 4 || 1 || 5 || 2 || 6 || 3 | ||
+ | |- | ||
+ | ! 5 || 0 || 5 || 3 || 1 || 6 || 4 || 4 | ||
+ | |- | ||
+ | ! 6 || 0 || 6 || 5 || 4 || 3 || 2 || 5 | ||
+ | |} | ||
+ | |} | ||
+ | |||
'''Theorem''': <math>\,\!F_P </math> for <math>p>1</math> is a field ''iff'' <small>([http://en.wikipedia.org/wiki/If_and_only_if if and only if])</small> <math>p</math> is a prime number | '''Theorem''': <math>\,\!F_P </math> for <math>p>1</math> is a field ''iff'' <small>([http://en.wikipedia.org/wiki/If_and_only_if if and only if])</small> <math>p</math> is a prime number | ||
Revision as of 22:52, 15 September 2009
The real numbers A set with two binary operators and two special elements s.t.
- Note: or means inclusive or in math.
Definition: A field is a set F with two binary operators : F×F → F, : F×F → F and two elements s.t.
Examples
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- is not a field (counterexample)
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Theorem: for is a field iff (if and only if) is a prime number
Tedious Theorem
- "cancellation property"
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