1 | initial version |

Note that the parent of `6/3`

is the `Rational Field`

:

```
sage: a = 6/3
sage: a.parent()
Rational Field
```

So, when you write `(6/3).is_prime()`

, you ask whether `6/3`

is prime as a rational number, see the documentation:

```
sage: a.is_prime?
A *prime* element is a non-zero, non-unit element p such that,
whenever p divides ab for some a and b, then p divides a or p
divides b.
```

So, since `6/3`

is a unit in `QQ`

, the answer should be `False`

```
sage: a.is_unit()
True
sage: a.is_prime()
False
```

2 | No.2 Revision |

Note that the parent of `6/3`

is the `Rational Field`

:

```
sage: a = 6/3
sage: a.parent()
Rational Field
```

So, when you write `(6/3).is_prime()`

, you ask whether `6/3`

is prime as a rational number, not as an integer, see the documentation:

```
sage: a.is_prime?
A *prime* element is a non-zero, non-unit element p such that,
whenever p divides ab for some a and b, then p divides a or p
divides b.
```

So, since `6/3`

is a unit in `QQ`

, the answer should be `False`

```
sage: a.is_unit()
True
sage: a.is_prime()
False
```

To see if `6/3`

is prime as an integer, just do:

```
sage: ZZ(6/3).is_prime()
True
```

3 | No.3 Revision |

Note that the parent of `6/3`

is the `Rational Field`

:

```
sage: a = 6/3
sage: a.parent()
Rational Field
```

So, when you write `(6/3).is_prime()`

, you ask whether `6/3`

is prime *as a rational number, number,* not as an integer, see the documentation:

```
sage: a.is_prime?
A *prime* element is a non-zero, non-unit element p such that,
whenever p divides ab for some a and b, then p divides a or p
divides b.
```

So, since `6/3`

is a unit in `QQ`

, the answer should be `False`

```
sage: a.is_unit()
True
sage: a.is_prime()
False
```

To see if `6/3`

is prime as an integer, just do:

```
sage: ZZ(6/3).is_prime()
True
```

4 | No.4 Revision |

Note that the parent of `6/3`

is the `Rational Field`

:

```
sage: a = 6/3
sage: a.parent()
Rational Field
```

So, when you write `(6/3).is_prime()`

, you ask whether `6/3`

is prime *as a rational number,* not as an integer, see the documentation:

```
sage: a.is_prime?
A *prime* element is a non-zero, non-unit element p such that,
whenever p divides ab for some a and b, then p divides a or p
divides b.
```

So, since `6/3`

is a unit in `QQ`

, the answer should be `False`

```
sage: a.is_unit()
True
sage: a.is_prime()
False
```

To see if `6/3`

is prime as an integer, just do:

```
sage: ZZ(6/3).is_prime()
True
```

So, it is very important in mathematics and in Sage to know where your elements are living. For example the polynomial `x^2-2`

can not be factorized in $\mathbb{Q}[x]$, but it does in $\overline{\mathbb{Q}}[x]$:

```
sage: x = polygen(QQ)
sage: (x^2-2).is_irreducible()
True
sage: x = polygen(QQbar)
sage: (x^2-2).is_irreducible()
False
```

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