To check that `A`

and `B`

belong to the field `F`

:

```
sage: A in F
True
sage: B in F
True
sage: A in F and B in F
True
```

Elements of `F`

are polynomials in the generator `a`

with coefficients in `GF(3)`

.
Here `a`

satisfies the relation `a^3 + a + 1 == 0`

due to the choice of `modulus`

in the definition of `F`

. So, to get all elements of `F`

it suffices to take polynomials of degree less than the degree of the modulus, i.e. of degree less than 3. (This is because any higher degree polynomial in `a`

can be simplified to one of degree < 3, using the relation `a^3 + a + 1 == 0`

.) This explains why you only get (all) polynomials of degree less than 3 in `a`

when you `enumerate(F)`

.

Since `A`

and `B`

are *defined* as polynomials in `a`

, they automatically belong to the field `F`

. You can even define, say, `C = a^5`

, and it will also belong to `F`

. Using the relation that `a`

satisfies you can see that `C == a^2 + a + 1`

.

Changing the numbers doesn't make your question any clearer. Please clarify with words what you want to know / what you are asking for.

what is the sage command to check A and B belongs to the field F.