Characteristic of F

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Characteristic of F

Theorem

Let $\mathbb{F}$ and $\mathbb{F}'$ be finite fields of size $q=p^{n}$ . Then $\mathbb{F}\cong\mathbb{F}'$ i.e. $\mathbb{F}$ and $\mathbb{F}'$ are isomorphic.

Definition (Minimal polynomial)

Let $\mathbb{F}$ be any field containing subfield $K$ . Given $\alpha\in\mathbb{F}$ , $\exists!$ minimal polynomial, $m(x)\in K[x]$ s.t. $m(\alpha)=0$ $m$ is unique and irreducible.

Lemma (Min poly. dividing poly. of same root)

Let $\mathbb{F}$ be a field containing field $K$ . Suppose $[\mathbb{F}:K]=n$ (i.e. the dimension of $\mathbb{F}$ , the vector space is $n$ ). Let $\alpha\in\mathbb{F}$ , $\alpha\not=0$ . Suppose $m(x)$ is the of $\alpha$ . If $f(x)\in K[x]$ is s.t. $f(\alpha)=0$ . Then $m(x)|f(x)$ in $K[x]$ .

Theorem (Characteristic of $\mathbb{F}$ )

If $\mathbb{F}$ is a finite field, then it must have a characteristic $p$ for some prime $p$ . The collection of elements $\{j \cdot 1 : 1 \le j \le p\}$ is isomorphic to $\mathbb{F}_{p}$ and is a subfield of $\mathbb{F}$ . We may therefore view $\mathbb{F}$ as a vector space over $\mathbb{F}_{p}$ which must necessarily have finite dimension. In other words, any finite field $\mathbb{F}$ must have cardinality $p^{n}$ for some prime $p$ and some natural number $n$ .