- #1
PsychonautQQ
- 784
- 10
Homework Statement
Describe the splitting field of x^9-1 over Z_13.
Homework Equations
The Attempt at a Solution
Well, seeing if any other of {1,...,12} would be some tricky arithmetic. Is my best bet here moving forward to just divide x^9-1 by x-1 since I know 1 is a root and then go from there?
So basically I'm looking for elements in Z*_13 with orders of 1,3 or 9. But there will be no element of order 9 because 9 does not divide 12 ( the order of Z*_13) so I'm looking for all elements of order 3. This makes checking to see if {1,...,12} work much easier.
Also, there since 3 is prime there is a subgroup of order 3, so there are at least 2 elements such that x^3=1 (mod 13). I have found these elements, 3 and 9, therefore I have 3 out of the 9 roots. What do I do from here? Divide x^9-1 by (x-1)(x-3)(x-9) and then divide the quotient by (x-r)? Is there a simpler way?
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