From the field $\{0,1,A,B\}$ I know that, by exhaustion, $A\cdot B=1$ but how do I get $A \cdot A $ and $B\cdot B$? How can I get the sum table? We haven't been thought almost anything about fields in class yet so I am stuck even with the other answers on the internet. I need a full explanation. I don't know what cyclic means or anything pretty much about fields.
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1The multiplicative group of a finite field is cyclic – Hagen von Eitzen Sep 23 '16 at 11:05
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There is a field of order 4 by using the irreducible quadratic $x^2+x+1$ [over integers mod 2]. Look up "finite fields" somewhere... – coffeemath Sep 23 '16 at 11:06
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5This has been handled many times. Here is perhaps the highest voted version. You may also find this helpful, and more recently we had this thread. That last item may be close to what you need. – Jyrki Lahtonen Sep 23 '16 at 11:07
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Note that $B=A+1$. Hence if you already know $AB=1$, we conclude $AA=A(B-1)=AB-A=1-A=1+A=B$ and $BB=(A+1)B=AB+B=1+B=2+A=A$.
Hagen von Eitzen
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You are missing at least the derivations of $B=A+1$ and $1+1=0$ (compare to the most recent reincarnation of this. Of course, you are free to make assumptions about what the OP knows already. – Jyrki Lahtonen Sep 23 '16 at 11:14
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1OP knows nothing about this. How do you get the derivation of $B = A + 1$? Also, it seems that $A\cdot A$ can't be $A$. Can you show this with multiplication only and not knowing $A+A$, $A+B$ or $B+B$ – The Bosco Sep 23 '16 at 11:16