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Thread: Algebraic structures - help with prove or disprove question

  1. #1
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    Algebraic structures - help with prove or disprove question

    Hi.
    I need to prove or disprove the following:

    1. Let $\displaystyle B=\left \{ s+tx|s,t\in \mathbb{Q} \right \}$ and $\displaystyle A=B\setminus \left \{ 0+0x \right \}$. A is a Group.
    2. Let $\displaystyle B=\left \{ s+tx|s,t\in \mathbb{Z}_5 \right \}$ and $\displaystyle A=B\setminus \left \{ 0+0x \right \}$. A is a Group.

    both with the operation:
    $\displaystyle (s+tx)*(u+vx)=(su+2vt)+(sv+tu)x$

    My intuition says 1 is true and 2 is false (the unit element in 1 is $\displaystyle (1+0\cdot x)=1$, and with simple algebra I can find the inverse element for a given element $\displaystyle (s+tx)\in A$), but I'm not sure about 2.
    Can someone please give me some help with it?

    Thanks in advanced!
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  2. #2
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    Re: Algebraic structures - help with prove or disprove question

    Hi Stormey,
    Do you know about field extensions? Your question is merely a disguised form of the following:
    Let F be a field and $\displaystyle x^2-2$ irreducible over F ($\displaystyle \sqrt2\notin F$). Then there is a field F' containing F and an element x in F' such that x2 = 2 and every element of F' is uniquely expressible as s + tx for some s and t in F. Of course the non-zero elements of F' form a group under multiplication. This is precisely your group in either 1 or 2.

    I really wouldn't want to prove the group axioms from scratch -- associativity would be a mess just from the definition of *. You said you could find the inverse of any element in case F = Q, but you didn't say what it is. Except for notation, this will also be the inverse when F is GF(5). For either case,
    $\displaystyle (a+bx)^{-1}=-a(2b^2-a^2)^{-1}+b(2b^2-a^2)^{-1}x$
    For a and b not both 0, $\displaystyle 2b^2-a^2\ne0$ since x^2-2 has no solution in F.
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  3. #3
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    Re: Algebraic structures - help with prove or disprove question

    Hi johng. thanks for your help.

    Eventually, this is (more or less) what I wrote... (this exercise was due to yesterday).
    I remembered something about B looks very much like the field $\displaystyle \mathbb{Q}(\sqrt 2)$, but I couldn't put my finger on why it is relevant.
    Hope to get some points for this question...
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