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Figure 1: extract from https://crypto.stanford.edu/pbc/notes/numbertheory/arith.html |
Now what is a field? Here's a definition from http://www.applet-magic.com/field.htm:
A fieldThe set ofconsists of a set of elements and two binary functions and defined on and whose values are in Both binary functions are associative, is equal to for all , and in and likewise for . The binary function , called addition is necessarily commutative; i.e., is equal to for all and in . The other function , called multiplication, is not necessarily commutative. There is an identity for each functions; i.e. there exist such that for all in and an such that for all in . There exist additive inverses for all elements of and multiplicative inverses for all elements of except the additive identity. This means that for any element in there exist and such that . Such is denoted as . For any element in that is not there exists and such that is equal to The element is usually denoted as . Furthermore there is distributivity of multiplication with respect to addition; i.e. is equal to for all , and in .
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Figure 2: mapping of elements of mod 7 under multiplication by 2 |
Figure 3 shows the full multiplication table:
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Figure 3: multiplication table for mod 7 |
Of course, as stated above there is no multiplicative inverse for the additive identity element. In the case of the mod 7, this element is 0 and it means we can't divide by this element just as in regular arithmetic. Figure 4 shows the full division table:
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Figure 4: division table for mod 7 with case of 5 divided by 3 highlighted |
Figure 5 explains the condition for division to be possible in a modular system:
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Figure 4: extract from https://crypto.stanford.edu/pbc/notes/numbertheory/arith.html |
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