In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod ${\displaystyle p}$ cohomology.

For a given prime number ${\displaystyle p}$, the Steenrod algebra ${\displaystyle A_{p}}$ is the graded Hopf algebra over the field ${\displaystyle \mathbb {F} _{p}}$ of order ${\displaystyle p}$, consisting of all stable cohomology operations for mod ${\displaystyle p}$ cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for ${\displaystyle p=2}$, and by the Steenrod reduced ${\displaystyle p}$th powers introduced in Steenrod (1953a, 1953b) and the Bockstein homomorphism for ${\displaystyle p>2}$.

The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.

A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring ${\displaystyle R}$, the cup product squaring operation yields a family of cohomology operations:

Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below.

These operations do not commute with suspension—that is, they are unstable. (This is because if ${\displaystyle Y}$ is a suspension of a space ${\displaystyle X}$, the cup product on the cohomology of ${\displaystyle Y}$ is trivial.) Steenrod constructed stable operations

for all ${\displaystyle i}$ greater than zero. The notation ${\displaystyle Sq}$ and their name, the Steenrod squares, comes from the fact that ${\displaystyle Sq^{n}}$ restricted to classes of degree ${\displaystyle n}$ is the cup square. There are analogous operations for odd primary coefficients, usually denoted ${\displaystyle P^{i}}$ and called the reduced ${\displaystyle p}$-th power operations:

The ${\displaystyle Sq^{i}}$ generate a connected graded algebra over ${\displaystyle \mathbb {Z} /2}$, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case ${\displaystyle p>2}$, the mod ${\displaystyle p}$ Steenrod algebra is generated by the ${\displaystyle P^{i}}$ and the Bockstein operation ${\displaystyle \beta }$ associated to the short exact sequence