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Theorem:

Let F be a finite field of characteristic p.

Then p is a prime and $\left | F \right |=p^{n},\left [ \exists n>0 \right ] \in \mathbb{N}$

Note that the characteristic of F denoted Char\left ( F \right )=p \neq 0

$\mathbf{For\space each \space \lambda \in \mathbb{Z}_{p} \space and \space a\in F}$

then $\lambda\cdot a=a+\cdot \cdot \cdot +a$ and $\lambda \cdot a \in F$ by definition of ring.

The ring axiom implies that F is a vector space over $\mathbb{Z}_{p}$. Since F is finite, F has finite dimension and so have a finite basis $b=\left \{ \vec{e}_{1},\cdot \cdot \cdot \vec{e}_{n} \right \}$

In a vector space, every vector element \alpha is a unique linear-combination of basis vector so $\forall \alpha \in F$, $\alpha =p_{1}\vec{e}_{1}+\cdot \cdot \cdot +p_{n}\vec{e}_{n} \forall \left \{ p_{i} \right \}^{n}_{i=1}$

so, $\mathbf{the \space size \space of \space F \space is\space p^{n}}$.

Two questions:

What is the motivation behind the first highlighted bold?

Secondly, is $\left | F \right |=p^{n}$ to be interpreted as $\left | F \right |=n\cdot p$ ?

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1 Answers1

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I guess $Z_p$ plays role of scalars in the definition of vector spaces, while the elements of $F$ are vectors. $\lambda \cdot a$ is just a notation for multiplication of a vector $a$ by a scalar $\lambda$.

Note also that the finite field of order $p^n$ can be constructed using an irreducible polynomial $p(x)$ of order $n$ with coefficients in $Z_p$. So we can treat the finite field of order $p^n$ as a set of polynomials in $Z_p[x]$ with order between $0$ and $n-1$, consider them as vectors. Then elements of $Z_p$ are scalars. Multiplication and addition are performed mod $p(x)$.

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