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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 can be simplified to one of degree < 3, using the relation a^3 + a + 1 == 0.) This explains why you only get 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.

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.

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 . Using the relation that a satisfies you can see that C == a^2 + a + 1.