I am a beginner in the filed theory. So, I have a few questions.
Consider a matrix $A$ whose each element is in $GF(2)$ (i.e. $0$ or $1$) and a matrix $B$, each element of which is of $512$ bits. If I multiply them together, $C = AB$, will the elements of $C$ be of $512$ bits, or just $0$ and $1$?
Similarly, if we consider the elements of $A$ is in $GF(2^{10})$, what would be the elements of $C$?
I am a bit confused. I think for the first question the answer should be $512$ bits and for the second question, the answer should also be the same. Can someone clarify if I am right or not?
English languages uses 26 alphabets. But when we read it we have to read them word by word (rather than individual letters). Analogously all Galois Fields $GF(2^n)$ for all $n$ use binary digits 0 and 1 at a primitive level. But words here are "n-bits at a time". So Galois Fields $GF(512)$ and $GF(1024)$ are to be understood as using same alphabet but their word lengths are 9 and 10 respectively.
The theory tells us how to add two $n$-bit string to fetch an $n$-bit string as their sum (bitwise XOR). And also the mathematicians have a rule (somewhat complicated) that specifies how to multiply two $n$-bit strings to yield another $n$-bit string as their product. The "sum" and "product" operations on binary strings obeying the usual conditions of our familiar number system (commutativity, assocoitivity, distributivity etc).