Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

splitting of primes in extension fields

splitting of primes in extension fields

Hi, I'm working on the maximal real subfield of the cyclotomic field of order 148. Let's call it L at the moment. Field L has quartic subfield K4. I want to see how a prime ideal I of K4 splits in the top field L. The following is the sage code I tried.

 K.<a>=CyclotomicField(148);

g=a+a**-1;

L.<b>=NumberField(g.minpoly());

M=L.subfields();

 K4=M[3][0];

 K4   K4

Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333. 333.

I have a fractional ideal I of K4 and would like to see how it splits in top field L. I tried the following way. way.

O=L.ring_of_integers();

J=I.O;

But it showed the error - unable to convert the fractional ideal I to Number Field L.

Could you please help me with this? Thank you.

splitting of primes in extension fields

Hi, I'm working on Call L the maximal real subfield of the cyclotomic field of order 148. Let's 148.

The field L has a quartic subfield — call it L at the moment. Field L has quartic subfield K4. K.

I want to see how a prime fractional ideal I of K4 K splits in the top field L. The following is the sage code L.

I tried. defined L and K as follows.

K.<a>=CyclotomicField(148);
sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly())
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K   g=a+a**-1;

L.<b>=NumberField(g.minpoly());

M=L.subfields();

K4=M[3][0];

K4

Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333.333

Then I defined a fractional ideal I in K.

I have a fractional ideal I of K4 and would like to To see how it splits in top field L. L, I tried the following way.

O=L.ring_of_integers();

J=I.O;
sage: O = L.ring_of_integers()
sage: J = I.O

But it showed the error - that gave an error: unable to convert the fractional ideal I to Number Field L.

Could you please help me with this? Thank you.

splitting of primes in extension fields

Call L the maximal real subfield of the cyclotomic field of order 148.

The field L has a quartic subfield — call it K.

I want to see how a fractional ideal I prime ideal of K splits in the top field L.

I defined L and K as follows.

sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly())
NumberField(g.minpoly()) 
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K  
Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
sage: I=K.ideal(37)
sage: F=I.factor()
sage: (Fractional ideal (37, b3))^4
sage : list=list(F)
sage: P=list[F]
sage: P
(Fractional ideal (37, b3), 4)
sage: P1= P[0];P1
Fractional ideal (37, b3)

Then I defined a fractional ideal I in K.

To I would like to see how it splits in L, the prime ideal P1 of K factor in the top field L. I tried the following way.code.

sage: O = L.ring_of_integers()
sage: J = I.O
O.P1

But that gave an I got the error: unable to convert the fractional ideal I to No compatible natural embeddings found for Number Field L.in b with defining polynomial x^36 - 37x^34 + 629x^32 - 6512x^30 + 45880x^28 - 232841x^26 + 878787x^24 - 2510820x^22 + 5476185x^20 - 9126975x^18 + 11560835x^16 - 10994920x^14 + 7696444x^12 - 3848222x^10 + 1314610x^8 - 286824x^6 + 35853x^4 - 2109x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37x^2 + 333

Could you please help me with this? Thank you.

splitting of primes in extension fields

Call L the maximal real subfield of the cyclotomic field of order 148.

The field L has a quartic subfield — call it K.

I want to see how a prime ideal of K splits in the top field L.

I defined L and K as follows.

sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly()) 
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K  
Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
sage: I=K.ideal(37)
sage: F=I.factor()
sage: (Fractional ideal (37, b3))^4
sage : list=list(F)
sage: P=list[F]
sage: P
(Fractional ideal (37, b3), 4)
sage: P1= P[0];P1
Fractional ideal (37, b3)

I would like to see how the prime ideal P1 of K factor in the top field L. I tried the following code.

sage: O = L.ring_of_integers()
sage: J = O.P1

But I got the error: No compatible natural embeddings found for Number Field in b with defining polynomial x^36 - 37x^34 + 629x^32 - 6512x^30 + 45880x^28 - 232841x^26 + 878787x^24 - 2510820x^22 + 5476185x^20 - 9126975x^18 + 11560835x^16 - 10994920x^14 + 7696444x^12 - 3848222x^10 + 1314610x^8 - 286824x^6 + 35853x^4 - 2109x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37x^2 + 333

Could you please help me with this? Thank you.

splitting of primes in extension fields

Call L the maximal real subfield of the cyclotomic field of order 148.

The field L has a quartic subfield — call it K.

I want to see how a prime ideal of K splits in the top field L.

I defined L and K as follows.

sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly()) 
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K  
Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
sage: I=K.ideal(37)
sage: F=I.factor()
sage: (Fractional ideal (37, b3))^4
sage: P=list[F]
sage: P
(Fractional ideal (37, b3), 4)
sage: P1= P[0];P1
Fractional ideal (37, b3)

I would like to see how the prime ideal P1 of K factor in the top field L. I tried the following code.

sage: O = L.ring_of_integers()
sage: J = O.P1

But I got the error: No compatible natural embeddings found for Number Field in b with defining polynomial x^36 - 37x^34 + 629x^32 - 6512x^30 + 45880x^28 - 232841x^26 + 878787x^24 - 2510820x^22 + 5476185x^20 - 9126975x^18 + 11560835x^16 - 10994920x^14 + 7696444x^12 - 3848222x^10 + 1314610x^8 - 286824x^6 + 35853x^4 - 2109x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37x^2 + 333

Could you please help me with this? Thank you.