1 | initial version |

You should define `g`

as an element of the univariate polynomial ring with coefficients in `F`

:

```
sage: R.<y>=F[]
sage: R
Univariate Polynomial Ring in y over Number Field in a with defining polynomial x^2 + 4
sage: g=y^4+16
sage: g.parent()
Univariate Polynomial Ring in y over Number Field in a with defining polynomial x^2 + 4
sage: g.is_irreducible()
False
sage: g.factor()
(y^2 - 2*a) * (y^2 + 2*a)
```

2 | No.2 Revision |

You should define `g`

as an element of the univariate polynomial ring with coefficients in `F`

:

`sage: `~~R.<y>=F[]
~~R.<y> = F[]
sage: R
Univariate Polynomial Ring in y over Number Field in a with defining polynomial x^2 + 4
sage: ~~g=y^4+16
~~g = y^4+16
sage: g.parent()
Univariate Polynomial Ring in y over Number Field in a with defining polynomial x^2 + 4
sage: g.is_irreducible()
False
sage: g.factor()
(y^2 - 2*a) * (y^2 + 2*a)

**EDIT** If you want the splitting field of `g`

, together with a map to embed `F`

into it, you can do:

```
sage: g.splitting_field('b')
Number Field in b with defining polynomial y^4 + 1
sage: g.splitting_field('b', map=True)
(Number Field in b with defining polynomial y^4 + 1, Ring morphism:
From: Number Field in a with defining polynomial x^2 + 4
To: Number Field in b with defining polynomial y^4 + 1
Defn: a |--> 2*b^2)
```

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