#Help with fields! Abstract algebra

31 messages · Page 1 of 1 (latest)

midnight marsh
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So far, I figured out (x)(x) = 1, but I cannot for the life of me figure out how to prove x + x = 1. I used math stack exchange (https://math.stackexchange.com/questions/168137/in-a-field-f-0-1-x-x-x-1-and-x-cdot-x-1) but I cannot make sense of it. Any help?

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Mathematics Stack Exchange
In a field $F=\{0,1,x\}$, $x + x = 1$ and $x\cdot x = 1$

Looking for some pointers on how to approach this problem:
Let $F$ be a field consisting of exactly three elements $0$, $1$, $x$. Prove that
$x + x = 1$ and that $x x = 1$.

ivory crescentBOT
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silk pecan
midnight marsh
silk pecan
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let me ask again

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what about the article?

midnight marsh
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ohhh

silk pecan
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since you quite explicitly are talking about... the article

midnight marsh
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I know I know

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it's just that I had this same problem in my HW and found that for help

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but

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In OP's conclusion, they stated this

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And I got confused in that last parentheses

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I don't understand their thought process

silk pecan
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Well $x+x=x$ means $x=0$, which is false

hidden sparrowBOT
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Omegabet_

186109587366215691.png
silk pecan
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$x+1$ cant be $1$ (else $x=0$), nor $x$ (else $1=0$), so $x+1=0$

hidden sparrowBOT
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Omegabet_

186109587366215691.png
silk pecan
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So if $x+x=0$, then $x+x=x+1$, which means $x=1$

hidden sparrowBOT
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Omegabet_

186109587366215691.png
silk pecan
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thus by exhaustion $x+x=1$

hidden sparrowBOT
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Omegabet_

186109587366215691.png
midnight marsh
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ohhhhhhhhhhhhh

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wow i feel dumb now

silk pecan
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which of course makes sense, take F_3 = Z/3Z to be the field of 3 elements

midnight marsh
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i didn't realized that what they had done was taken that x+1=0 is true and apply it to the x+x portion

silk pecan
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x is then the element dubbed 2, and 2+2=4=1 mod 3

midnight marsh
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wow okay

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thank you so much

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+close