1 | initial version |

Are the ramified primes the ones which divide the field discriminant? If this is so you can get them by

```
sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
sage: [x[0] for x in list(K.discriminant().factor())]
[5, 41]
```

2 | No.2 Revision |

Are the ramified primes the ones which divide the field discriminant? If this is so you can get them by

`sage: K.<y> = NumberField(x^4 - `~~420*x^2 ~~x^2 + ~~40000)
~~1)
sage: [x[0] for x in list(K.discriminant().factor())]
~~[5, 41]
~~[2, 3]

If splitting means that the prime factors then you can check this like this:

```
sage: is_split = lambda F,x:sum([t[1] for t in list(F.factor(x))])>1
```

for example:

```
sage: K.<y> = NumberField(x^2 + 1)
sage: for x in range(30):
if is_prime(x):
print x%4,is_split(K,x)
....:
2 True
3 False
1 True
3 False
3 False
1 True
1 True
3 False
3 False
1 True
```

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