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Group rings

  1. Nov 2, 2007 #1
    I have about 5 questions all of a similar form .....

    [tex] \mathbb{F}[C_2] \cong \mathbb{F} \times \mahbb{F} [/tex]

    if 1+1 [tex] \neq 0 in \mathbb{F} [/tex]

    [tex]\Re [C_3] \cong \Re \times C [/tex]

    [tex] \Re [C_4] \cong \Re \times \Re \times C [/tex]

    these were on the first sheet given out and I still don't know how to do them after 4 weeks, any help on how to do them??
     
  2. jcsd
  3. Nov 3, 2007 #2

    HallsofIvy

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    "Do them"? Do you mean prove them? Start by writing out the definitions.
    (I would have thought that [itex]\mathbb{F}[/itex] was a field but apparently not.)

    What are C2, C3, C4?
     
  4. Nov 3, 2007 #3

    morphism

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    I guess C_n is the cyclic group of order n. But still, what the OP posted doesn't make a lot of sense, e.g. what's the distinction between F and [itex]\mathbb{F}[/itex]? Also, is [itex]\Re[/itex] actually [itex]\mathbb{R}[/itex]? (The former is the "real part" of a complex number, the latter is the set of real numbers.) And, in the last two, what is C? Finally, in what context are the direct products being taken?
     
  5. Nov 8, 2007 #4
    obviously i haven't made myself clear, i'll post the question in it's entirety...

    Let A,B be algebras over a field F. We say that A and B are isomorphic over F written [tex] A\cong_F B [/tex] when there exists a bijective ring homomorphism [tex] \varphi : A \rightarrow B [/tex] which is also linear over F, i.e. satisfies [tex] \\\varphi(a\lambda) = \varphi(a) \lambda \mbox{ for all a} \in A \ \lambda \in F [/tex]

    show that i) [tex] F[C_2] \cong_F F \times F [/tex] if 1+1 not equal 0 in F
    ii) [tex] R[C_3] \cong_R R \times C [/tex]

    iii) [tex] \\ R[C_4] \cong_R R \times R \times C [/tex]

    iv) [tex]\\ R[C_6] \cong R \times R \times C \times C [/tex]


    here R = field of real numbers, C = field of complex numbers, F is an arbitrary field although in the question it is constrained by the given condition. Also all C_n's are cyclics and by F[C_n] we mean the group ring of C_n over F, that is the ring whose elements are linear combinations of the group elements with coefficients in F.
     
  6. Dec 21, 2007 #5

    morphism

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    For some reason this problem popped into my head today, so I gave it another go and managed to solve it. There are two things we need:
    (1) [itex]F[C_n] \cong_F F[x]/(x^n - 1)[/itex]. This can be easily proved using the evaluation homomorphism at a generator of C_n.
    (2) The Chinese remainder theorem for rings, and the observation that a collection of maximal ideals is pairwise comaximal. (The CRT is something that slipped my mind at the time.)

    So now we have:
    [tex]F[C_2] \cong F[x]/(x^2 - 1) \cong F[x]/(x+1) \times F[x]/(x-1) \cong F \times F[/tex]

    [tex]\mathbb{R}[C_3] \cong \mathbb{R}[x]/(x^3 - 1) \cong \mathbb{R}[x]/(x-1) \times \mathbb{R}[x]/(x^2 + x + 1) \cong \mathbb{R} \times \mathbb{C}[/tex]

    [tex]\mathbb{R}[C_4] \cong \mathbb{R}[x]/(x^4 - 1) \cong \mathbb{R}[x]/(x-1) \times \mathbb{R}[x]/(x+1) \times \mathbb{R}[x]/(x^2 + 1) \cong \mathbb{R} \times \mathbb{R} \times \mathbb{C}[/tex]

    [tex]\mathbb{R}[C_6] \cong \mathbb{R}[x]/(x^6 - 1) \cong \mathbb{R}[x]/(x-1) \times \mathbb{R}[x]/(x+1) \times \mathbb{R}[x]/(x^2+x+1) \times \mathbb{R}[x]/(x^2-x+1) \cong \mathbb{R} \times \mathbb{R} \times \mathbb{C} \times \mathbb{C}[/tex]

    Better late than never I suppose!
     
    Last edited: Dec 21, 2007
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