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Math Help - Field extension

  1. #1
    Newbie dangkhoa's Avatar
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    Field extension

    Let K(s) be the field of rational functions in one variable s, i.e the field of fractions of K[s]. 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)

    How do you prove it?

    Thank you very much
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  2. #2
    Senior Member
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    Quote Originally Posted by dangkhoa View Post
    Let K(s) be the field of rational functions in one variable s, i.e the field of fractions of K[s]. 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)

    How do you prove it?

    Thank you very much
    Let F=K(s^n). Then, \{1_{F}, s, s^2, \cdots, s^{n-1}\} is a basis of vector space K(s) over F=K(s^n), which implies that every element of K(s) can be written uniquely in the form of c_0 + c_1s \cdots c_{n-1}s^{n-1} (c_i \in F). Thus [K(s):K(s^n)] = n.

    f(t)=t^n-s^n \in K(s^n)[t] is an irreducible monic polynomial of degree n over K(s^n) such that f(s)=0.
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