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







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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















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







share|cite|improve this question






















  • 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













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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 21 at 12:44









primer

1




1












  • 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


















  • 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
















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















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