We can assume that p ≠ 2 , 3 because 0 is a root of...

Lorena Beard

Lorena Beard

Answered

2022-06-29

We can assume that p 2 , 3 because 0 is a root of f ( x ) in F 2 and F 3 . Let Q be the "squares subgroup" (?) of index 2 in F p . If 2 and 3 are not in Q (this means that ( x 2 2 ) ( x 2 3 ) doesn't have roots in F p ), then 2 3 Q (?), and then x 2 6 has a root in F p

Answer & Explanation

Karla Hull

Karla Hull

Expert

2022-06-30Added 20 answers

By the squares subgroup, they mean the subgroup
Q = { a F p a = b 2  for some  b F p } .
In other words, Q is the image of the homomorphism φ : F p F p given as x x 2
In the below, I am also going to assume that p { 2 , 3 }
Since ker φ = { 1 , 1 } and 1 1 (why?), we see that G has index 2 in F p
Now, if 2 , 3 , 6 Q, then this means that the cosets 2 Q , 3 Q , 6 Q are all not equal to Q. Since Q has index 2 in F p , this means that 2 Q = 3 Q = 6 Q
Thus, one of 2 , 3 , 6 must be in Q. In turn, the given polynomial has a root in F p (in fact, in F p , if p 2 , 3)

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