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Thread: additive and multiplicative groups of a field are not isomorphic

  1. #1
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    additive and multiplicative groups of a field are not isomorphic

    This is from Dummit and Foote Abstract Algebra.

    Prove that the additive and multiplicative groups of a field are never isomorphic.

    Hint from the book:

    Consider three cases: when $\displaystyle |\mathbb{F}|$ is finite, when $\displaystyle -1 \ne 1$ and when $\displaystyle -1=-1$ in $\displaystyle \mathbb{F}$.

    The finite part is not bad.

    let $\displaystyle |\mathbb{F}|=n$

    then $\displaystyle |(\mathbb{F},+)|=n$ but $\displaystyle |(\mathbb{F}, \cdot )|=n-1$

    because 0 does not have an inverse in the multiplicative group.

    For the next case I think contradiction is the way to go so I supposed that there exists an isomorphism $\displaystyle \phi : (\mathbb{F},\cdot) \to (\mathbb{F},+)$ and that $\displaystyle -1 \ne 1$

    I know that $\displaystyle \phi(1)=0$ becuase the identity must map to the identity. Also that

    $\displaystyle \phi(1)=\phi[(-1)(-1)]=\phi(-1)+\phi(-1)=0$

    and for any $\displaystyle a \in \mathbb{F}$

    $\displaystyle \phi[(-1)(a)]=\phi(-1)+\phi(a)$
    $\displaystyle \phi(1(-a))=\phi(1)+\phi(-a)$

    I know that inverses must get mapped to inverses by an isomorphism.

    So I don't see how to get a contradiction So I know I am missing something.

    Thanks

    TES
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    Quote Originally Posted by TheEmptySet View Post
    This is from Dummit and Foote Abstract Algebra.

    Prove that the additive and multiplicative groups of a field are never isomorphic.

    Hint from the book:

    Consider three cases: when $\displaystyle |\mathbb{F}|$ is finite, when $\displaystyle -1 \ne 1$ and when $\displaystyle -1=-1$ in $\displaystyle \mathbb{F}$.

    The finite part is not bad.

    let $\displaystyle |\mathbb{F}|=n$

    then $\displaystyle |(\mathbb{F},+)|=n$ but $\displaystyle |(\mathbb{F}, \cdot )|=n-1$

    because 0 does not have an inverse in the multiplicative group.

    For the next case I think contradiction is the way to go so I supposed that there exists an isomorphism $\displaystyle \phi : (\mathbb{F},\cdot) \to (\mathbb{F},+)$ and that $\displaystyle -1 \ne 1$

    I know that $\displaystyle \phi(1)=0$ becuase the identity must map to the identity. Also that

    $\displaystyle \phi(1)=\phi[(-1)(-1)]=\phi(-1)+\phi(-1)=0$

    and for any $\displaystyle a \in \mathbb{F}$

    $\displaystyle \phi[(-1)(a)]=\phi(-1)+\phi(a)$
    $\displaystyle \phi(1(-a))=\phi(1)+\phi(-a)$

    I know that inverses must get mapped to inverses by an isomorphism.

    So I don't see how to get a contradiction So I know I am missing something.

    Thanks

    TES
    suppose there's a group isomorphism $\displaystyle f: (F,+) \longrightarrow F^{\times}.$

    1) $\displaystyle \text{char} F = 2$: then $\displaystyle (f(1))^2=f(2)=f(0)=1.$ the equation $\displaystyle x^2=1$ in a field of characteristic 2 has the unique solution x = 1. thus $\displaystyle f(1)=1=f(0),$ which is impossible because $\displaystyle f$ is injective.

    2) $\displaystyle \text{char} F \neq 2$: since $\displaystyle f$ is surjective, there exists $\displaystyle a \in F: \ f(a)=-1.$ but then $\displaystyle f(0)=1=(f(a))^2=f(2a).$ thus $\displaystyle 2a=0,$ and hence $\displaystyle a=0.$ but then $\displaystyle -1=f(a)=f(0)=1.$ contradiction!
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