Answer by rburing for I would like to generate elements of finite filed $GF(3^{15})$. To do so I have used the following code : F.<x> = GF(3^15) for i in range(3^15): print i,"=>", F.fetch_int(i) But this code failed to generate elements, occurring errors. On there hand the above code works fine for $GF(3^{2})$ given below: F.<x> = GF(3^2) for i in range(3^2): print i,"=>", F.fetch_int(i) Produces: 0 => 0 1 => 1 2 => 2 3 => x 4 => x + 1 5 => x + 2 6 => 2*x 7 => 2*x + 1 8 => 2*x + 2. Where is the problem?

See the [documentation of GF](http://doc.sagemath.org/html/en/reference/finite_rings/sage/rings/finite_rings/finite_field_constructor.html); the different implementations depending on the size are the first thing mentioned. In particular, $\log_3(2^{16}) \approx 10.09$ means `GF(3^2), ..., GF(3^10)` use the Givaro implementation while `GF(3^11), GF(3^12), ...` use the PARI implementation. The `fetch_int` method is not available in the PARI implementation, which explains the error. Note that the mapping you are interested in is very simple: for example $7$ is $2\cdot 3 + 1$, so its ternary (base 3) digits are `21`, and the corresponding element of `F. = GF(3^k)` is `2*x + 1` ($3$ is replaced by $x$). Since `F` is a vector space over `GF(3)` with basis `1`, `x`, `x^2`, ..., `x^(k-1)`, a vector (or list) of elements of `GF(3)` can be converted into `F`. For example, `F([1,2])` yields `2*x + 1`; the reason for the "reversal" is that the "digits" are interpreted in [little endian](https://en.wikipedia.org/wiki/Endianness) order. This is convenient because the output of e.g. `7.digits(3)` is also in little endian order, yielding `[1,2]`. We can put this together: F. = GF(3^15) for i in srange(F.cardinality()): print(i, '=>', F(i.digits(3))) Though it is kind of pointless to print that gigantic list when you already understand the mapping. Also, I suppose that a `fetch_int` method could be added to the PARI implementation for convenience.