1 | initial version |

When defining a finite field of non-prime order, it is useful to give a name to the generator.

Likewise, when defining a polynomial ring, it is useful to give a name to the polynomial variable.

In the example, `GF(2^8, 'a')`

returns the finite field with $2^8$
elements, with `a`

as the display name of its generator.

And `R.<x> = K[]`

simultaneously defines `R`

as a polynomial ring
over the field `K`

, and definds `x`

as its generator, i.e., the
polynomial variable.

So `x^254`

is the monic monomial of degree 254 in this polynomial ring.

2 | No.2 Revision |

When defining a finite field of non-prime order, it is useful to give a name to the generator.

Likewise, when defining a polynomial ring, it is useful to give a name to the polynomial variable.

In the example, `GF(2^8, 'a')`

returns the finite field with $2^8$
elements, with `a`

as the display name of its generator.

And `R.<x> = K[]`

simultaneously defines `R`

as a polynomial ring
over the field `K`

, with a polynomial variable that displays as 'x',
and ~~definds ~~defines `x`

as its generator, i.e., ~~the
~~the polynomial variable.

So `x^254`

is the monic monomial of degree 254 in this polynomial ring.

For more, read the documentation or/and the source code for `GF`

:

```
sage: GF?
sage: GF??
```

and for `PolynomialRing`

:

```
sage: PolynomialRing?
sage: PolynomialRing??
```

Note that `R.<x> = K[]`

is transformed by the Sage preparser into:

```
sage: preparse("R.<x> = K[]")
"R = K['x']; (x,) = R._first_ngens(1)"
```

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