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Math Help - galois theory

  1. #1
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    galois theory

    Can someone please help me with this question finding it very difficult

    Thanks Edgar


    Let K be a field and s an indeterminate. Then K(s) is a field extension of K(s^n). Prove that [K(s) : K(s^n)] = n.
    Hence show that the minimum polynomial of s over K(s^n) is t^n - s^n.

    Hint: first show that s satisifies a polynomial of degree n over K(s^n); this gives <=. Then show that {1,s,...,s^(n-1)} is linearly independent over K(s^n); this gives you >=.
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  2. #2
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    Quote Originally Posted by edgar davids View Post
    Can someone please help me with this question finding it very difficult

    Thanks Edgar


    Let K be a field and s an indeterminate. Then K(s) is a field extension of K(s^n). Prove that [K(s) : K(s^n)] = n.
    Hence show that the minimum polynomial of s over K(s^n) is t^n - s^n.

    Hint: first show that s satisifies a polynomial of degree n over K(s^n); this gives <=. Then show that {1,s,...,s^(n-1)} is linearly independent over K(s^n); this gives you >=.
    Let f(x) = x^n - s^n \in K(s^n) then clearly f(s) = 0 so if p(x) is the minimal polynomial for s over K(s^n) we have that p(x) | f(x) \implies k= \deg p(x) \leq \deg f(x) = n \implies k\leq n. Now that means that \{ 1, s,s^2, ... ,s^{k-1} \} is a basis for K(s) over K(s^n). If we can show that \{1,s,s^2 , ... s^{n-1} \} is linearly independent then it would means k\geq n because the dimension of a linearly independent set cannot the dimension of a basis. To show that \{1,s,s^2,...,s^{n-1} \} is linearly independent over K(s^n) it is equivalent to showing that no element in this set can be expressed as a linear combination of the other elements. Take for example, 1, we cannot express 1 as a linear combination a_1 s+a_2s^n + ... + a_{n-1}s^{n-1} where a_i \in K(s^n) because a_i = b^{(i)}_0 + b^{(i)}_1 s^n + b^{(i)}_2 s^{2n} + ... so it is impossible to get intermediately exponents between 1 and n. Thus, k=n and this means p(x) = x^n - s^n.

    This is Mine 87th Post!!!
    Last edited by ThePerfectHacker; February 14th 2008 at 08:40 PM.
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