Difference between revisions of "09-240/Classnotes for Tuesday September 15"
From Drorbn
(→Examples: Addition and multiplication tables.) |
(→Examples: Expand, fix table format.) |
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# <math>\,\!F_2 = \{ 0, 1 \}</math> | # <math>\,\!F_2 = \{ 0, 1 \}</math> | ||
# <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 | + | # <math>\,\!F_6 = \{ 0, 1,2,3,4,5 \}</math> is not a field because not every element has a multiplicative inverse. |
+ | #: Let <math>a = 2.</math> | ||
+ | #: Then <math>a \cdot 0 = 0, a \cdot 1 = 2, a \cdot 3 = 0, a \cdot 4 = 2, a \cdot 5 = 4</math> | ||
+ | #: Therefore F4 fails; there is '''no''' number ''b'' in ''F''<sub>6</sub> s.t. ''a · b'' = 1 | ||
{| | {| | ||
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! + !! 0 !! 1 | ! + !! 0 !! 1 | ||
|- | |- | ||
− | ! 0 | + | ! 0 |
+ | | 0 || 1 | ||
|- | |- | ||
− | ! 1 | + | ! 1 |
+ | | 1 || 0 | ||
|- | |- | ||
|} | |} | ||
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! × !! 0 !! 1 | ! × !! 0 !! 1 | ||
|- | |- | ||
− | ! 0 | + | ! 0 |
+ | | 0 || 0 | ||
|- | |- | ||
− | ! 1 | + | ! 1 |
+ | | 0 || 1 | ||
|} | |} | ||
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! + !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 | ! + !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 | ||
|- | |- | ||
− | ! 0 | + | ! 0 |
+ | | 0 || 1 || 2 || 3 || 4 || 5 || 6 | ||
|- | |- | ||
− | ! 1 | + | ! 1 |
+ | | 1 || 2 || 3 || 4 || 5 || 6 || 0 | ||
|- | |- | ||
− | ! 2 | + | ! 2 |
+ | | 2 || 3 || 4 || 5 || 6 || 0 || 1 | ||
|- | |- | ||
− | ! 3 | + | ! 3 |
+ | | 3 || 4 || 5 || 6 || 0 || 1 || 2 | ||
|- | |- | ||
− | ! 4 | + | ! 4 |
+ | | 4 || 5 || 6 || 0 || 1 || 2 || 3 | ||
|- | |- | ||
− | ! 5 | + | ! 5 |
+ | | 5 || 6 || 0 || 1 || 2 || 3 || 4 | ||
|- | |- | ||
− | ! 6 | + | ! 6 |
+ | | 6 || 0 || 1 || 2 || 3 || 4 || 5 | ||
|} | |} | ||
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! × !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 | ! × !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 | ||
|- | |- | ||
− | ! 0 | + | ! 0 |
+ | | 0 || 0 || 0 || 0 || 0 || 0 || 0 | ||
|- | |- | ||
− | ! 1 | + | ! 1 |
+ | | 0 || 1 || 2 || 3 || 4 || 5 || 0 | ||
|- | |- | ||
− | ! 2 | + | ! 2 |
+ | | 0 || 2 || 4 || 6 || 1 || 3 || 1 | ||
|- | |- | ||
− | ! 3 | + | ! 3 |
+ | | 0 || 3 || 6 || 2 || 5 || 1 || 2 | ||
|- | |- | ||
− | ! 4 | + | ! 4 |
+ | | 0 || 4 || 1 || 5 || 2 || 6 || 3 | ||
|- | |- | ||
− | ! 5 | + | ! 5 |
+ | | 0 || 5 || 3 || 1 || 6 || 4 || 4 | ||
|- | |- | ||
− | ! 6 | + | ! 6 |
+ | | 0 || 6 || 5 || 4 || 3 || 2 || 5 | ||
|} | |} | ||
|} | |} |
Revision as of 23:19, 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 because not every element has a multiplicative inverse.
- Let
- Then
- Therefore F4 fails; there is no number b in F6 s.t. a · b = 1
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Theorem: for is a field iff (if and only if) is a prime number
Tedious Theorem
- "cancellation property"
...