Re: [GAP Forum] Quadratic residue in finite field

[Stefan Kohl]

On Tue, February 7, 2017 10:21 am, Surinder Kaur wrote:
> If F is a field with q = p^m -1 elements where m is even and p is an odd
> prime,

Since the existence of a field F with q elements implies that q is a prime power
and since for an odd prime p we have q = p^m - 1 even, we can conclude that
q is in fact a power of 2, and that F has characteristic 2.

> A the set of all quadratic residues and B = {4 b - 1 | b is
> primitive element of F}.

Since F has characteristic 2, we have B = {1}.

> Then my aim is to prove A intersection B is non
> empty i.e. for some primitive element b the element 4b-1 is a square of
> some element.

Just take any primitive element b. -- Then we have 4b-1 = 1 (since F has
characteristic 2), and 1 is the square of itself. qed.

-- Does this help you?

Best regards,

Stefan Kohl

P.S.: The GAP Forum is intended for questions on GAP.
  -- General questions on mathematics are more on-topic on Math.SE.




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Re: [GAP Forum] Quadratic residue in finite field

[Surinder Kaur]

If F is a field with q = p^m -1 elements where m is even and p is an odd
prime, A the set of all quadratic residues and B = {4 b - 1 | b is
primitive element of F} . Then my aim is to prove A intersection B is non
empty i.e. for some primitive element b the element 4b-1 is a square of
some element. How  to check whether the conjecture holds atleast for some p
and m's.
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Re: [GAP Forum] Quadratic residue in finite field

[Bill Allombert]

On Tue, Feb 07, 2017 at 01:40:47PM +0530, Surinder Kaur wrote:
> How can one check whether an element in a finite field with p^m elements is
> a quadratic residue or not?

Just test whether
a^((p^m-1)/2) = 1

Cheers,
Bill.

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