GAP command
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From: galoisgroupquestion on 6 May 2010 23:35 Hi, I need to verify some of my Galois group exercises using GAP. How do I find the Galois group of the polynomial x^4 + 2x^2 + x +3 over Q using GAP? Can I also find the splitting field of it using GAP? How do I also find the Galois group of the same polynomial over GF(7) rather than Q? I googled and tried it, but it was not successful so far. Any help will be appreciated. Thanks.
From: Risto Kauppila on 7 May 2010 10:17 07.05.2010 10:35, galoisgroupquestion kirjoitti: > Hi, I need to verify some of my Galois group exercises using GAP. > > How do I find the Galois group of the polynomial > x^4 + 2x^2 + x +3 over Q using GAP? > > Can I also find the splitting field of it using GAP? > > How do I also find the Galois group of the same polynomial over GF(7) rather than Q? > > I googled and tried it, but it was not successful so far. > > Any help will be appreciated. > > Thanks. Listen some good reggae (for example Dr Mooch: Virginia) and think the problem anew. Rike
From: galoisgroupquestion on 7 May 2010 13:43 > 07.05.2010 10:35, galoisgroupquestion kirjoitti: > > Hi, I need to verify some of my Galois group > exercises using GAP. > > > > How do I find the Galois group of the polynomial > > x^4 + 2x^2 + x +3 over Q using GAP? > > > > Can I also find the splitting field of it using > GAP? > > > > How do I also find the Galois group of the same > polynomial over GF(7) rather than Q? > > > > I googled and tried it, but it was not successful > so far. > > > > Any help will be appreciated. > > > > Thanks. > > Listen some good reggae (for example Dr Mooch: > Virginia) and > think the problem anew. > > Rike Don't reply if you have no intention to help. It is a waste of time to read your stupid reply indeed.
From: Chip Eastham on 7 May 2010 18:36 On May 7, 5:43 pm, galoisgroupquestion <galoisgroupquest...(a)hotmail.com> wrote: > > 07.05.2010 10:35, galoisgroupquestion kirjoitti: > > > Hi, I need to verify some of my Galois group > > exercises using GAP. > > > > How do I find the Galois group of the polynomial > > > x^4 + 2x^2 + x +3 over Q using GAP? > > > > Can I also find the splitting field of it using > > GAP? > > > > How do I also find the Galois group of the same > > polynomial over GF(7) rather than Q? > > > > I googled and tried it, but it was not successful > > so far. > > > > Any help will be appreciated. > > > > Thanks. > > > Listen some good reggae (for example Dr Mooch: > > Virginia) and > > think the problem anew. > > > Rike > > Don't reply if you have no intention to help. > It is a waste of time to read your stupid reply indeed. I like Rike's suggestion very much and appreciate the time taken to reply. However I will suggest that while the polynomial x^4 + 2x^2 + x + 3 is irreducible over Q, it is not irreducible over Z/7Z. I believe the GAP command you want requires an irreducible polynomial as input. See: http://www.gap-system.org/Manuals/doc/htm/ref/CHAP056.htm regards, chip
From: Gerry on 10 May 2010 10:04
On May 8, 12:36 am, Chip Eastham <hardm...(a)gmail.com> wrote: > On May 7, 5:43 pm, galoisgroupquestion > > > > > > <galoisgroupquest...(a)hotmail.com> wrote: > > > 07.05.2010 10:35, galoisgroupquestion kirjoitti: > > > > Hi, I need to verify some of my Galois group > > > exercises using GAP. > > > > > How do I find the Galois group of the polynomial > > > > x^4 + 2x^2 + x +3 over Q using GAP? > > > > > Can I also find the splitting field of it using > > > GAP? > > > > > How do I also find the Galois group of the same > > > polynomial over GF(7) rather than Q? > > > > > I googled and tried it, but it was not successful > > > so far. > > > > > Any help will be appreciated. > > > > > Thanks. > > > > Listen some good reggae (for example Dr Mooch: > > > Virginia) and > > > think the problem anew. > > > > Rike > > > Don't reply if you have no intention to help. > > It is a waste of time to read your stupid reply indeed. > > I like Rike's suggestion very much and appreciate > the time taken to reply. > > However I will suggest that while the polynomial > x^4 + 2x^2 + x + 3 is irreducible over Q, it is > not irreducible over Z/7Z. I believe the GAP > command you want requires an irreducible polynomial > as input. See: > > http://www.gap-system.org/Manuals/doc/htm/ref/CHAP056.htm > > regards, chip- Hide quoted text - > > - Show quoted text - Use p1:=X^4+2*X^2+X+3; g1:=Galois(p1); gives "S4" The roots are : x1= 1/2*sqrt(r)-1/2*sqrt(-4-r-2/sqrt(r)) x2= 1/2*sqrt(r)+1/2*sqrt(-4-r-2/sqrt(r)) x3=-1/2*sqrt(r)-1/2*sqrt(-4-r+2/sqrt(r)) x4=-1/2*sqrt(r)+1/2*sqrt(-4-r+2/sqrt(r)) and s = 1/2(-389+3*I*sqrt(11631)) r = -4/3 + 40/(3*s^(1/3))+1/3*s^(1/3) So s is a root of the quadratic : x^2+389*x+64000 and r is a root of the cubic : x^3+4*x^2-8*x-1 i don't know the answer to yout other questions. |