$(2)-$ cyclotomic cosets modulo a prime
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Let $p$ be an odd prime. Assume $2$ is a quadratic residue modulo $p$. Is it true that the $(2)-$ cyclotomic cosets modulo $p$ are ${{0}}, {{Q}}, {{N}}$, where $Q$ are the quadratic residues modulo $p$ and $N$ are the non-residues?
My thoughts;
I looked at the $(2)$ - cyclotomic coset modulo $p$ containing $1$, which is always a quadratic residue. The coset then consists of successive powers of $2$ modulo $p$. I can't see why these are all the quadratic residues, though?
abstract-algebra number-theory information-theory quadratic-residues
asked Dec 3 '18 at 18:39
the manthe man
721715
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add a comment |
$begingroup$
Let $p$ be an odd prime. Assume $2$ is a quadratic residue modulo $p$. Is it true that the $(2)-$ cyclotomic cosets modulo $p$ are ${{0}}, {{Q}}, {{N}}$, where $Q$ are the quadratic residues modulo $p$ and $N$ are the non-residues?
My thoughts;
I looked at the $(2)$ - cyclotomic coset modulo $p$ containing $1$, which is always a quadratic residue. The coset then consists of successive powers of $2$ modulo $p$. I can't see why these are all the quadratic residues, though?
abstract-algebra number-theory information-theory quadratic-residues
asked Dec 3 '18 at 18:39
the manthe man
721715
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What do you mean by $(2)$-cyclotomic coset?
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– Lukas Kofler
Dec 3 '18 at 18:41
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The $(2)$ cyclotomic coset containing $i$ is ${{i2^k : k geq 0}}.$
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– the man
Dec 3 '18 at 19:38
add a comment |
$begingroup$
Let $p$ be an odd prime. Assume $2$ is a quadratic residue modulo $p$. Is it true that the $(2)-$ cyclotomic cosets modulo $p$ are ${{0}}, {{Q}}, {{N}}$, where $Q$ are the quadratic residues modulo $p$ and $N$ are the non-residues?
My thoughts;
I looked at the $(2)$ - cyclotomic coset modulo $p$ containing $1$, which is always a quadratic residue. The coset then consists of successive powers of $2$ modulo $p$. I can't see why these are all the quadratic residues, though?
abstract-algebra number-theory information-theory quadratic-residues
asked Dec 3 '18 at 18:39
the manthe man
721715
$endgroup$
Let $p$ be an odd prime. Assume $2$ is a quadratic residue modulo $p$. Is it true that the $(2)-$ cyclotomic cosets modulo $p$ are ${{0}}, {{Q}}, {{N}}$, where $Q$ are the quadratic residues modulo $p$ and $N$ are the non-residues?
My thoughts;
I looked at the $(2)$ - cyclotomic coset modulo $p$ containing $1$, which is always a quadratic residue. The coset then consists of successive powers of $2$ modulo $p$. I can't see why these are all the quadratic residues, though?
abstract-algebra number-theory information-theory quadratic-residues
abstract-algebra number-theory information-theory quadratic-residues
asked Dec 3 '18 at 18:39
the manthe man
721715
asked Dec 3 '18 at 18:39
the manthe man
721715
asked Dec 3 '18 at 18:39
the manthe man
721715
asked Dec 3 '18 at 18:39
the manthe man
721715
asked Dec 3 '18 at 18:39
the manthe man
721715
721715
$begingroup$
What do you mean by $(2)$-cyclotomic coset?
$endgroup$
– Lukas Kofler
Dec 3 '18 at 18:41
$begingroup$
The $(2)$ cyclotomic coset containing $i$ is ${{i2^k : k geq 0}}.$
$endgroup$
– the man
Dec 3 '18 at 19:38
add a comment |
$begingroup$
What do you mean by $(2)$-cyclotomic coset?
$endgroup$
– Lukas Kofler
Dec 3 '18 at 18:41
$begingroup$
The $(2)$ cyclotomic coset containing $i$ is ${{i2^k : k geq 0}}.$
$endgroup$
– the man
Dec 3 '18 at 19:38
$begingroup$
What do you mean by $(2)$-cyclotomic coset?
$endgroup$
– Lukas Kofler
Dec 3 '18 at 18:41
$begingroup$
What do you mean by $(2)$-cyclotomic coset?
$endgroup$
– Lukas Kofler
Dec 3 '18 at 18:41
$begingroup$
The $(2)$ cyclotomic coset containing $i$ is ${{i2^k : k geq 0}}.$
$endgroup$
– the man
Dec 3 '18 at 19:38
$begingroup$
The $(2)$ cyclotomic coset containing $i$ is ${{i2^k : k geq 0}}.$
$endgroup$
– the man
Dec 3 '18 at 19:38
add a comment |
1 Answer
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You're right, the maximal size of such a coset is roughly $log_2 p$ rounded up but the number of quadratic residues is roughly $p/2.$ Now take $p$ large enough.
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
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add a comment |
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$begingroup$
You're right, the maximal size of such a coset is roughly $log_2 p$ rounded up but the number of quadratic residues is roughly $p/2.$ Now take $p$ large enough.
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
$endgroup$
add a comment |
$begingroup$
You're right, the maximal size of such a coset is roughly $log_2 p$ rounded up but the number of quadratic residues is roughly $p/2.$ Now take $p$ large enough.
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
$endgroup$
add a comment |
$begingroup$
You're right, the maximal size of such a coset is roughly $log_2 p$ rounded up but the number of quadratic residues is roughly $p/2.$ Now take $p$ large enough.
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
$endgroup$
You're right, the maximal size of such a coset is roughly $log_2 p$ rounded up but the number of quadratic residues is roughly $p/2.$ Now take $p$ large enough.
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
answered Dec 5 '18 at 21:43
kodlukodlu
3,390716
3,390716
add a comment |
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$begingroup$
What do you mean by $(2)$-cyclotomic coset?
$endgroup$
– Lukas Kofler
Dec 3 '18 at 18:41
$begingroup$
The $(2)$ cyclotomic coset containing $i$ is ${{i2^k : k geq 0}}.$
$endgroup$
– the man
Dec 3 '18 at 19:38