- #1
Rectifier
Gold Member
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The problem
Consider a ring ##(Z_{12}, \oplus, \otimes)##
Find ##5^{-1}##.
The attempt
I am basically searching for an inverse to ##5^{-1}## (if I am correct then judging from previous examples in my book, 5 wouldn't qualify)
Since rings are about residues (the set ##Z_{12}## includes residues as integer elements from 0 to 11) we are basically searching for an integer ##x## that when multiplied by ##5^{-1}## and for which ## R_{12}(x \cdot 5^{-1}) = 1 ## so if ##x =65## would produce that result sine ##65/5=13## and the residue when dividing by ##12## is ##1##, ##65## is an inverse to ##a## since its residue product is ##1##.
Now, for some reason we are supposed to search for ##gcd(12, 5)## I don't really understand why and why there is 5 there and not ##5^{-1}## (well, I get that ##5^{-1}## is less between 0 and 1, thus it is not included since it is not an integer but why is it 5?). Somehow they know that the integer is going to be 1 so they want to express that integer as a Bezout's identity (Euclides algorithm backwards) and finally get an expression for (something). I have not figured it out yet since I don't know why we don't take ##x## from before into consideration yet.
I hope that this was not too long and not many questions. Please help, I have really been trying to figure this stuff out but got stuck at this.
Consider a ring ##(Z_{12}, \oplus, \otimes)##
Find ##5^{-1}##.
The attempt
I am basically searching for an inverse to ##5^{-1}## (if I am correct then judging from previous examples in my book, 5 wouldn't qualify)
Since rings are about residues (the set ##Z_{12}## includes residues as integer elements from 0 to 11) we are basically searching for an integer ##x## that when multiplied by ##5^{-1}## and for which ## R_{12}(x \cdot 5^{-1}) = 1 ## so if ##x =65## would produce that result sine ##65/5=13## and the residue when dividing by ##12## is ##1##, ##65## is an inverse to ##a## since its residue product is ##1##.
Now, for some reason we are supposed to search for ##gcd(12, 5)## I don't really understand why and why there is 5 there and not ##5^{-1}## (well, I get that ##5^{-1}## is less between 0 and 1, thus it is not included since it is not an integer but why is it 5?). Somehow they know that the integer is going to be 1 so they want to express that integer as a Bezout's identity (Euclides algorithm backwards) and finally get an expression for (something). I have not figured it out yet since I don't know why we don't take ##x## from before into consideration yet.
I hope that this was not too long and not many questions. Please help, I have really been trying to figure this stuff out but got stuck at this.