### 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 - 37*x^34 + 629*x^32 - 6512*x^30 + 45880*x^28 - 232841*x^26 + 878787*x^24 - 2510820*x^22 + 5476185*x^20 - 9126975*x^18 + 11560835*x^16 - 10994920*x^14 + 7696444*x^12 - 3848222*x^10 + 1314610*x^8 - 286824*x^6 + 35853*x^4 - 2109*x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333

Could you please help me with this? Thank you.