This book is about the mod 2 Steenrod algebra A2 and its action on the polynomial algebra P(n) = F2[x1, …, xn] in n variables, where F2 is the field of two elements. Polynomials are graded by degree, so that Pd(n) is the set of homogeneous polynomials of degree d. Although our subject has its origin in the work of Norman E. Steenrod in algebraic topology, we have taken an algebraic point of view. We have tried as far as possible to provide a self-contained treatment based on linear algebra and representations of finite matrix groups. In other words, the reader does not require knowledge of algebraic topology, although the subject has been developed by topologists and is motivated by problems in topology.

There are many bonuses for working with the prime p = 2. There are no coefficients to worry about, so that every polynomial can be written simply as a sum of monomials. We use a matrix-like array of 0s and 1s, which we call a ‘block’, to represent a monomial in P(n), where the rows of the block are formed by the reverse binary expansions of its exponents. Thus a polynomial is a set of blocks, and the sum of two polynomials is the symmetric difference of the corresponding sets. Using block notation, the action of A2 on P(n) can be encoded in computer algebra programs using standard routines on sets, lists and arrays. In addition, much of the literature on the Steenrod algebra and its applications in topology concentrates on the case p=2. Often a result for p=2 has later been extended to all primes, but there are some results where no odd prime analogue is known.

We begin in Chapter 1 with the algebra map Sq : P(n)→P(n) defined on the generators by Sq(xi) = xi +x2i. The map Sq is the total Steenrod squaring operation, and the Steenrod squares Sqk : Pd(n) → Pd+k(n) are its graded parts.