A problem in Dummit and Foote states: Let k be a field and let k(x) be the field of rational f

Tagiuraoob

Tagiuraoob

Answered question

2022-02-18

A problem in Dummit and Foote states:
Let k be a field and let k(x) be the field of rational functions in x with coefficients from k. Let tϵk(x) be the rational function P(x)Q(x) with relatively ' polynomials P(x),Q(x)ϵk[x], with Q(x)0. Then k(x) is an extension of k(t) and to compute its degree it is necessary to compute the minimal polynomial with coefficients in k(t) satisfied by x.
By k(t), do they mean k adjoin t, i.e. the set of polynomials k0+k1P(x)Q(x)++kn(P(x)Q(x))n? Or do they mean the set of rational functions in t, e.g. P(x)Q(x)+12(P(x)Q(x))2+3
Thanks for answer!

Answer & Explanation

Tye Rhodes

Tye Rhodes

Beginner2022-02-19Added 4 answers

By k(t), they mean the set of rational functions in t (the second option). These form a field. The other, which might be denoted k[t], do not form a field.

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