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Algebraic Extensions, vector spaces

  1. Feb 19, 2012 #1
    Suppose that F/K is an algebraic extension, S is a subset of F with S/K a vector space and [itex] s^n \in S [/itex] for all s in S. I want to show that if char(K) isn't 2, then S is a subfield of F.

    Since F/K is algebraic, we know that [itex]\text{span} \lbrace 1,s,s^2,...\rbrace [/itex] is a field for any s in S. Thus, I want to define [itex] E= \bigcup _{s \in S} \text{span} \lbrace 1,s,s^2,...\rbrace [/itex]. Since S/K is a v.s., we have [itex] \text{span}\lbrace 1,s,s^2,...\rbrace \subseteq S[/itex] and since E is just a union of subsets of S, [itex] E \subset S [/itex]. Also, by the way we defined E, [itex] S \subseteq E [/itex] so E = S.

    (E,+) is a group by the fact that S/K is a v.s. Each nonzero element has an inverse in S since [itex] \text{span} \lbrace 1,s,s^2,...\rbrace [/itex] is a field and each s belongs to one of those fileds (so that field would have the inverse of s).

    The hard part seems to be proving closure of multiplication. Suppose that [itex] s_1 s_2 = s + v [/itex] for [itex] s_1,s_2,s\in S [/itex] and [itex] v\in F [/itex]. We have already shown that S contains all inverses, so [itex] s_2 = s_1^{-1} s + s_1^{-1} v \in S [/itex]. Whence, [itex] s_1^{-1} v \in S [/itex]. Obviously, I want to show that [itex] v\in S [/itex], but progress has stopped at this point.

    Any suggestions would be appreciated.
     
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  3. Feb 19, 2012 #2

    morphism

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    Think about how you can get an expression involving [itex]s_1+s_2[/itex] and [itex]s_1s_2[/itex].
     
  4. Feb 19, 2012 #3
    Since S is a v.s., we get [itex]s_1+s_2[/itex] by vector addition.

    For [itex] s_1 s_2[/itex] you have to multiply two linear combos in S (whose basis must be a subset of the basis for F/K). If the product of two basis elements of S is again in S, then the product of any two elements of S would be in S.

    However, I do not see how to show this.
     
  5. Feb 19, 2012 #4

    morphism

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    I'm not sure how to phrase my hint differently, so I apologize if this is giving away too much: Consider [itex](s_1+s_2)^2[/itex].
     
  6. Feb 19, 2012 #5
    Yay. That makes sense why char(K) can't be 2 then.

    Thanks a lot... why couldn't I think of that.
     
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