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Math Help - subfield

  1. #1
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    subfield

    Let K be a field an let L be a finite field extension of K.
    Let K be a subset of R and R a subset of L, so that for all a,b in R a+b, a*b in R.
    Show: R is a subfield of L.

    I still have to show the existence of inverse elements of the multiplication. The rest is all clear. But I have absolutely no real idea...

    Maybe a proof by contradiction???
    Can I somehow use that the field extension is algebraic?
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  2. #2
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    Quote Originally Posted by Icarus0 View Post
    Let K be a field an let L be a finite field extension of K.
    Let K be a subset of R and R a subset of L, so that for all a,b in R a+b, a*b in R.
    Show: R is a subfield of L.

    I still have to show the existence of inverse elements of the multiplication. The rest is all clear. But I have absolutely no real idea...

    Maybe a proof by contradiction???
    Can I somehow use that the field extension is algebraic?
    every element of R is algebraic over K. now look at the minimal polynomial of a non-zero element of R over K.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    every element of R is algebraic over K. now look at the minimal polynomial of a non-zero element of R over K.
    What has this to do with multiplicatice inverse elements? The minimal polynomial of a in R is just a irreducible polynomial f in K[X] with f(a)=0 ????
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  4. #4
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    Quote Originally Posted by Icarus0 View Post
    What has this to do with multiplicatice inverse elements? The minimal polynomial of a in R is just a irreducible polynomial f in K[X] with f(a)=0 ????
    let f(x)=x^n + c_1x^{n-1} + \cdots + c_{n-1}x + c_n \in K[x] be the minimal polynomial of 0 \neq a \in R. then c_n \neq 0 and a^n + c_1a^{n-1} + \cdots + c_{n-1}a + c_n=0. now let b=-c_n^{-1}(a^{n-1} + c_1a^{n-2} + \cdots + c_{n-1}).

    it's clear that b \in R and ab=1.
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