SOLUTION: University of Nairobi Modern Algebra Polynomial Equation Homework 6

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MAT 150C: MODERN ALGEBRA Homeachievement 6 Instructions. Please transcribe the retort to each substance, including the computational ones, in connected sentences and interpret your achievement. Just the retort (amend or not) is not plenty. Transcribe your seduce in perfect page and upadvise to Gradescope after a while the amend orientation. Make unmistakable to evidence to Gradescope which pages suit to each substance. Finally, if you used another sources or discussed the substance after a while classmates, be unmistakable to retain it in your homework. 1. Let R be a limited (purport that it has a limited enumerate of elements) integral inclosure. Show that R is a field. 2. (a) Let f (x) ∈ R[x]. Assume z ∈ C is a radicle of f . Show that z is a radicle of f as polite, where z indicates the complicated conjugate. (b) Deduce that perfect uncongenial polynomial in R[x] has extent 1 or 2. Give a compulsory and sufficient condition for the polynomial ax2 + bx + c ∈ R[x] (a 6= 0) to be uncongenial in R[x]. (c) Let f (x) ∈ R[x] be a polynomial of odd extent. Show, after a whileout using calculus, that f has a radicle in R. 3. Let F ⊆ K be a scope production, and let α ∈ K be such that [F(α) : F] = 5. Show that F(α2 ) = F(α). For the present exercises, we indicate ζn := e2π √ −1/n , a old-fashioned n-rooth of 1 in C. 4. Assume p is excellent and r > 0. Show that [Q(ζpr ) : Q] = (p − 1)pr−1 . (Hint: you can appeal to previous homeworks). 1 5. Show that ζ5 6∈ Q(ζ7 ). 1 In open, [Q(ζn ) : Q] = Tot(n), where Tot is Euler’s totient part 1 ...
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