Power[ ] is not working inside of Sum [ ]?

Questions
,
1

Morning! I’m trying to solve expression, using palette comands only. I want to get all roots of expression, than get sum of squared roots. But ^2 is not working at all. What am I doing wrong?
My code be like:

a = Solve[x^7-1==0,x];

res = \!\(UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(7\)]\(Power[\((x /. a[\([n]\)])\), 2]\)\)

Also is there any way to simplify result? FullSimplify returns 0 and Simplify is not working at all. Res is:

1-(-1)^(1/7)+(-1)^(2/7)-(-1)^(3/7)+(-1)^(4/7)-(-1)^(5/7)+(-1)^(6/7)

2

There are 7 solutions for x they are

10855Capture.JPG
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.
.Using N[a]
you get

{{x → 1.},

{x → -0.9009688679024191 - 0.4338837391175581 I},

{x →
0.6234898018587336 +
0.7818314824680298 I},

{x → -0.22252093395631445 -
0.9749279121818236 I},

{x → -0.22252093395631434 +
0.9749279121818236 I},

{x →
0.6234898018587335 -
0.7818314824680299 I},

{x → -0.900968867902419 +
0.43388373911755823 I}}

3

Yes, and I’m trying to get Sum of all roots^2

photo_2020-12-14_15-17-48.jpg
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But as you can see on my screenshot, each root is not powered. And even after FullSimplity, roots are not 1 or -1, but (-1)^(1/7)

4

a = Solve[x^7 - 1 == 0, x]

Out[42]= {{x → 1}, {x → -(-1)^(1/7)}, {x → (-1)^(
2/7)}, {x → -(-1)^(3/7)}, {x → (-1)^(
4/7)}, {x → -(-1)^(5/7)}, {x → (-1)^(6/7)}}

  t = N[Total[x^2 /. a]]

Out[53]= 1.1102230246251565*10^-16 + 0. I

Chop[t]

Out[54]= 0

5

If you check e.g. the Wikipedia page for resultants you find that this sum-of-squared-roots can be found using the computation below.

In[810]:= Coefficient[Resultant[x^2 - y, x^7 - 1, x], y^6]

(* Out[810]= 0 *)

Also there is a more direct method using RootSum (which may well use the above approach under the hood).

In[815]:= RootSum[#^7 - 1 &, #^2 &]

(* Out[815]= 0 *)
6

Does anyone can explain, whyn am I getting 0 as a result of FullSimplify?

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7

The result is zero and FullSimplify manages to make that determination. This is a good outcome. If FullSimplify gave, say, Pi^2+EulerGamma`, that would indicate it was having a very bad day.

8

But as I can see, the result should be equal to 1. Or not?

9

I’d go with the “or not” option on that.