Quote:

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**dopi** can someone check my solutions please

f is poly t^5 -3 in Q[t], where alpha= 3^1/5 in R, epsilon = e^(2pi/5)

**(a) need to find zeros of f in terms of a and epsilon**

my solution to (a)

alpha, alpha*epsilon, alpha*epsilon^2, alpha*epsilon^3, alpha*epsilon^4

This is good. Quote:

**(b) L = (alpha, epsilon). want to show that L is a splitting field of f over Q.**

my solution to (b)

zeros of f are given by (t -alpha)*(t -alpha*epsilon)*(t -alpha*epsilon^2)*(t -alpha*epsilon^3)*(t -alpha*epsilon^4)

so L = (alpha, epsilon) is splitting field

This is good. Quote:

**(c) want to state min polynomial of alpha over Q ane epsilon over Q(alpha), and from this i want to write down the bases of the extensions Q(alpha):Q and L:Q(alpha) and then find the value of [L:Q]**

my solution to (c)

min poly of alpha t^5 - 3

min poly of epsilon t^4 + t^3 + t^3 + t^2 + t + 1

a basis for Q(alpha) over Q is {1, alpha, alpha^2, alpha^3,alpha^4}

a basis for Q(alpha, epsilon) over Q(alpha) is {1, epsilon, epsilon^2, epsilon^3}

therefore [L:Q] = 9

thank you

There is a problem here. (Surprised)