associates of primes in $ mathbb{Z}[tau]$
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In $mathbb{Z}[tau]$, for $tau = frac{1+sqrt{5}}{2},$ how do I input through GAP its primes and their respective associates? An example for their inert and split primes would be much appreciated. thanks!
abstract-algebra number-theory gap
asked Nov 21 at 12:44
primer
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up vote
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In $mathbb{Z}[tau]$, for $tau = frac{1+sqrt{5}}{2},$ how do I input through GAP its primes and their respective associates? An example for their inert and split primes would be much appreciated. thanks!
abstract-algebra number-theory gap
asked Nov 21 at 12:44
primer
1
I use magmaR<x> := PolynomialRing(Integers()); K<a> := NumberField(x^2-5); OK := Integers(K); tau := (a+1)/2; Factorization( 2*OK + 6*tau*OK); U,m:= UnitGroup(OK); m(2*U.1);
– reuns
Nov 21 at 13:24
Is your question how you identify, whether a given prime in Z is split?
– ahulpke
Nov 21 at 13:53
@ahulpke yes, sir. Also, I want to know how to input through GAP the associates of the primes. thanks
– primer
Nov 21 at 15:14
@ahulpke My goal is somewhat like this, saw this one at math.stackexchange.com/questions/2525037/… gap> tau:=(1+ER(5))/2; gap> pol:=MinimalPolynomial(Rationals,tau); gap> Factors(PolynomialModP(pol,7)); #for p = 7
– primer
Nov 21 at 15:28
I still struggle to understand what you mean by "input through GAP". Maybe you can describe in an example the information you would like GAP to give you.
– ahulpke
Nov 21 at 16:45
|
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In $mathbb{Z}[tau]$, for $tau = frac{1+sqrt{5}}{2},$ how do I input through GAP its primes and their respective associates? An example for their inert and split primes would be much appreciated. thanks!
abstract-algebra number-theory gap
asked Nov 21 at 12:44
primer
1
In $mathbb{Z}[tau]$, for $tau = frac{1+sqrt{5}}{2},$ how do I input through GAP its primes and their respective associates? An example for their inert and split primes would be much appreciated. thanks!
abstract-algebra number-theory gap
abstract-algebra number-theory gap
asked Nov 21 at 12:44
primer
1
asked Nov 21 at 12:44
primer
1
asked Nov 21 at 12:44
primer
1
asked Nov 21 at 12:44
primer
1
asked Nov 21 at 12:44
primer
1
1
I use magmaR<x> := PolynomialRing(Integers()); K<a> := NumberField(x^2-5); OK := Integers(K); tau := (a+1)/2; Factorization( 2*OK + 6*tau*OK); U,m:= UnitGroup(OK); m(2*U.1);
– reuns
Nov 21 at 13:24
Is your question how you identify, whether a given prime in Z is split?
– ahulpke
Nov 21 at 13:53
@ahulpke yes, sir. Also, I want to know how to input through GAP the associates of the primes. thanks
– primer
Nov 21 at 15:14
@ahulpke My goal is somewhat like this, saw this one at math.stackexchange.com/questions/2525037/… gap> tau:=(1+ER(5))/2; gap> pol:=MinimalPolynomial(Rationals,tau); gap> Factors(PolynomialModP(pol,7)); #for p = 7
– primer
Nov 21 at 15:28
I still struggle to understand what you mean by "input through GAP". Maybe you can describe in an example the information you would like GAP to give you.
– ahulpke
Nov 21 at 16:45
|
show 3 more comments
I use magmaR<x> := PolynomialRing(Integers()); K<a> := NumberField(x^2-5); OK := Integers(K); tau := (a+1)/2; Factorization( 2*OK + 6*tau*OK); U,m:= UnitGroup(OK); m(2*U.1);
– reuns
Nov 21 at 13:24
Is your question how you identify, whether a given prime in Z is split?
– ahulpke
Nov 21 at 13:53
@ahulpke yes, sir. Also, I want to know how to input through GAP the associates of the primes. thanks
– primer
Nov 21 at 15:14
@ahulpke My goal is somewhat like this, saw this one at math.stackexchange.com/questions/2525037/… gap> tau:=(1+ER(5))/2; gap> pol:=MinimalPolynomial(Rationals,tau); gap> Factors(PolynomialModP(pol,7)); #for p = 7
– primer
Nov 21 at 15:28
I still struggle to understand what you mean by "input through GAP". Maybe you can describe in an example the information you would like GAP to give you.
– ahulpke
Nov 21 at 16:45
I use magma
R<x> := PolynomialRing(Integers()); K<a> := NumberField(x^2-5); OK := Integers(K); tau := (a+1)/2; Factorization( 2*OK + 6*tau*OK); U,m:= UnitGroup(OK); m(2*U.1);– reuns
Nov 21 at 13:24
I use magma
R<x> := PolynomialRing(Integers()); K<a> := NumberField(x^2-5); OK := Integers(K); tau := (a+1)/2; Factorization( 2*OK + 6*tau*OK); U,m:= UnitGroup(OK); m(2*U.1);– reuns
Nov 21 at 13:24
Is your question how you identify, whether a given prime in Z is split?
– ahulpke
Nov 21 at 13:53
Is your question how you identify, whether a given prime in Z is split?
– ahulpke
Nov 21 at 13:53
@ahulpke yes, sir. Also, I want to know how to input through GAP the associates of the primes. thanks
– primer
Nov 21 at 15:14
@ahulpke yes, sir. Also, I want to know how to input through GAP the associates of the primes. thanks
– primer
Nov 21 at 15:14
@ahulpke My goal is somewhat like this, saw this one at math.stackexchange.com/questions/2525037/… gap> tau:=(1+ER(5))/2; gap> pol:=MinimalPolynomial(Rationals,tau); gap> Factors(PolynomialModP(pol,7)); #for p = 7
– primer
Nov 21 at 15:28
@ahulpke My goal is somewhat like this, saw this one at math.stackexchange.com/questions/2525037/… gap> tau:=(1+ER(5))/2; gap> pol:=MinimalPolynomial(Rationals,tau); gap> Factors(PolynomialModP(pol,7)); #for p = 7
– primer
Nov 21 at 15:28
I still struggle to understand what you mean by "input through GAP". Maybe you can describe in an example the information you would like GAP to give you.
– ahulpke
Nov 21 at 16:45
I still struggle to understand what you mean by "input through GAP". Maybe you can describe in an example the information you would like GAP to give you.
– ahulpke
Nov 21 at 16:45
|
show 3 more comments
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I use magma
R<x> := PolynomialRing(Integers()); K<a> := NumberField(x^2-5); OK := Integers(K); tau := (a+1)/2; Factorization( 2*OK + 6*tau*OK); U,m:= UnitGroup(OK); m(2*U.1);– reuns
Nov 21 at 13:24
Is your question how you identify, whether a given prime in Z is split?
– ahulpke
Nov 21 at 13:53
@ahulpke yes, sir. Also, I want to know how to input through GAP the associates of the primes. thanks
– primer
Nov 21 at 15:14
@ahulpke My goal is somewhat like this, saw this one at math.stackexchange.com/questions/2525037/… gap> tau:=(1+ER(5))/2; gap> pol:=MinimalPolynomial(Rationals,tau); gap> Factors(PolynomialModP(pol,7)); #for p = 7
– primer
Nov 21 at 15:28
I still struggle to understand what you mean by "input through GAP". Maybe you can describe in an example the information you would like GAP to give you.
– ahulpke
Nov 21 at 16:45