# Steenrod Operations

## `SteenrodOperation(f, i): RngMPolElt, RngIntElt -> RngMPolElt`

The $i$-th Steenrod operation $P^i(f)$ of $f$, which must be a multivariate polynomial with coefficients in a finite field, and $i$ must be a non-negative integer.

## `Example: Steenrod Operation (ex-dbbce3)`

We demonstrate an elementary use of Steenrod operations.

```magma
> K:=GF(3);
> F4:=MatrixGroup<4,K |
>   [-1,0,0,0, 1,1,0,0, 0,0,1,0, 0,0,0,1],
>   [1,1,0,0, 0,-1,0,0, 0,1,1,0, 0,0,0,1],
>   [1,0,0,0, 0,1,-1,0, 0,0,-1,0, 0,0,1,1],
>   [1,0,0,0, 0,1,0,0, 0,0,1,1, 0,0,0,-1] >;
> R := InvariantRing(F4);
> f2 := InvariantsOfDegree(R, 2)[1];
> f4 := SteenrodOperation(f2, 1);
> f10 := SteenrodOperation(f4, 3);
> f4;
2*x1^4 + x1^3*x3 + 2*x1^3*x4 + x1*x3^3 + 2*x1*x4^3 + 2*x2^3*x3 + x2^3*x4 +
    2*x2*x3^3 + x2*x4^3 + x4^4
> f10;
2*x1^10 + x1^9*x3 + 2*x1^9*x4 + x1*x3^9 + 2*x1*x4^9 + 2*x2^9*x3 + x2^9*x4 +
    2*x2*x3^9 + x2*x4^9 + x4^10
> f4 in R;
true
> f10 in R;
true

```