Let p and l be two distinct primes different from 2. Let Omega be an algebraic closure of F_p and let w in Omega be a primitive l-th root of unity. If x in F_l, the element w^x is well defined since w^l=1...

Laila Murphy

Laila Murphy

Answered question

2022-11-03

Question about The Quadratic Reciprocity Law Proof
In A Course in Arithmetic (Serre) he beings by proving the quadratic reciprocity law (Thereom 6) as follows:
Let p and l be two distinct primes different from 2. Let Ω be an algebraic closure of F p and let w Ω be a primitive l-th root of unity. If x F l , the element w x is well defined since w l = 1...
My question is as follows: I don't understand why w x is defined at all. I know that w n is defined if n is an integer but x is an element of the field F l . So what do we even mean by w x in a precise sense? Does it still reside in Ω? I can't think of a polynomial where one of its roots is w x .

Answer & Explanation

Maffei2el

Maffei2el

Beginner2022-11-04Added 20 answers

Step 1
The field F l is just Z / l Z , so an element of it is an equivalence class of integers modulo l. Since w l = 1, you can raise w to the power of any representative of the equivalence class and the outcome will not depend on the choice of representative.
Step 2
In other words, w x is defined as w n , where n is any integer in the equivalence class x.

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