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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.=CyclotomicField(148);

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;
 
g=a+a**-1;L.<b>=NumberField(g.minpoly());
 M=L.subfields();
 
L.=NumberField(g.minpoly());
 K4=M[3][0];
  
M=L.subfields();
 
K4=M[3][0];
 
 K4   Number Field in b3 with defining polynomial x^4 - 37*x^2 + 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();
 
O=L.ring_of_integers();
 
J=I.O;J=I.O;

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;

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.

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.

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.

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.