BCH code $F^{q}_{2}$ field

I'm trying to understand the algorithm of construction the bch code.

Polynomial $m_{\alpha}(x) \in F_{2}[x]$ is called the minimal polynomial for an element $\alpha \in F^{q}_{2}$ if it is an irreducible polynomial of minimal degree for which $\alpha$ is the root. In particular, the minimal polynomial for a primitive element $\alpha$ is called a primitive polynomial. It can be shown that the roots of the minimal polynomial $m_{\alpha}(x)$ are $\{\alpha, \alpha^2, \alpha^4 ... \alpha^{2^{s}}\}$. It can be shown that the polynomial $\prod\limits_{I=0}^{s}(x + \alpha^{2^I}) = x^{s+1} + \lambda_sx^s + ... +\lambda_1x + \lambda_0$ has coefficients from $F_{2}$ and is the minimal polynomial for $\alpha$, as well as for all elements of the field that together with $\alpha$ belong to the same cyclotomic class.

What elements does the $F^{q}_{2}$ contain? I thought that this elements are vectors of 0 and 1, but how it can be the root of polynomial.

Solution 1:

The elements of $F_2^s$ are objects of the form $b_1\alpha+b_2\alpha^2+\cdots+b_s\alpha^{2^s}$, where each $b_j\in\{0,1\}$. Of course any such element can be represented more concisely as a string of $s$ bits. But this field construction (which exploits the fact that $\alpha$ is a root of a polynomial) makes it easier to see how the sum and product of two such objects is computed. (Once those rules are understood, they can be carefully applied directly to the bit strings themselves without ever mentioning the field; but this is where the rules come from.)