1 | initial version |

First note that written as`-4536*sqrt(33)+61128`

, the number is part of Sage's
symbolic ring.

To work with algebraic numbers, the best is to work either
- in a number field,
- in the field of algebraic numbers, `QQbar`

),
- in the field of real algebraic numbers, `AA`

.

Example of working in `AA`

:

```
sage: a = -4536*sqrt(33)+61128
sage: aa = AA(a)
sage: aa
35070.66383530351?
sage: aa.sqrt()
187.2716311545973?
sage: bb = aa.sqrt()
sage: bb
187.2716311545973?
sage: bb.radical_expression()
42*sqrt(33) - 54
```

2 | No.2 Revision |

First note that written as`-4536*sqrt(33)+61128`

, the number is part of Sage's
symbolic ring.

To work with algebraic numbers, the best is to work ~~either
- in a number field,
- ~~either

- in a number field,
- in the field of algebraic numbers,
`QQbar`

~~), -~~), - in the field of real algebraic numbers,
`AA`

.

Example of working in `AA`

:

```
sage: a = -4536*sqrt(33)+61128
sage: aa = AA(a)
sage: aa
35070.66383530351?
sage: aa.sqrt()
187.2716311545973?
sage: bb = aa.sqrt()
sage: bb
187.2716311545973?
sage: bb.radical_expression()
42*sqrt(33) - 54
```

Example of working in a number field:

First we check the approximate value of sqrt(33):

```
sage: sqrt(33).n()
5.74456264653803
```

Create an embedded number field:

```
sage: K.<r33> = NumberField(x^2-33, 5.8)
```

Check if our number field element is a square:

```
sage: a = -4536*r33+61128
sage: a.is_square()
True
```

Take the square root:

```
sage: a.sqrt()
42*r33 - 54
```

3 | No.3 Revision |

First note that written as`-4536*sqrt(33)+61128`

, the number is part of Sage's
symbolic ring.

To work with algebraic numbers, the best is to work either

- in a number field,
- in the field of algebraic numbers,
`QQbar`

), - in the field of real algebraic numbers,
`AA`

.

Example of working in `AA`

:

```
sage: a = -4536*sqrt(33)+61128
sage: aa = AA(a)
sage: aa
35070.66383530351?
sage: aa.sqrt()
187.2716311545973?
sage: bb = aa.sqrt()
sage: bb
187.2716311545973?
sage: bb.radical_expression()
42*sqrt(33) - 54
```

Example of working in a number field:

First we check the approximate value of sqrt(33):

```
sage: sqrt(33).n()
5.74456264653803
```

Create an embedded number field:

```
sage: K.<r33> = NumberField(x^2-33, 5.8)
```

Check if our number field element is a square:

```
sage: a = -4536*r33+61128
sage: a.is_square()
True
```

Take the square root:

```
sage: a.sqrt()
42*r33 - 54
```

Final note: in all cases, the method `sqrt`

will return one of the two square roots.
Keep in mind that there is another one --- it's just the opposite of the one you got.

So if you let `b = a.sqrt()`

, then remember that `-b`

is also a square root of `a`

.

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