Some properties of a field

Properties of a (general) field $F$

Let $F$ be a field and $E/F$ a field extension.

  1. Commutative ring $R (\neq 0)$ is a field $\Leftrightarrow$ Ideals of $R$ are only $0\left(=(0)\right)$ or $R\left(=(1)\right)$
    • Proof. $(\Rightarrow):\,a\in I\Rightarrow 1= aa^{-1}\in I$. $(\Leftarrow):\,0\neq a\in R\Rightarrow 1\in R=(a)$.
    • Cor. Homomorphism from a field to a ring is injective.
  2. $F[x]$ is a p.i.d. (Principal Ideal Domain)
    • Proof. If ideal $I$ of $F[x]$ is not $0$, there exists a minimum (positive) degree polynomial.
  3. Suppose $u\in E$ is algebraic over $F$ and its minimum polynomial is $g(x)$. Then, if $g(x)$ is irreducible in $F[x]\Rightarrow F[u]$ is a field.
    • Proof. $F[x]/(g(x))\cong F[u]$. Any ideals of $F[x]/(g(x))$ can be represented as $I/(g(x))$ where $(g(x))\subset I$ is an ideal of $F[x]$. Hence $I=(f(x))$ and $f(x)\bigl| g(x)$. If $g(x)$ is irreducible, $\left(f(x)\right)=\left(1\right)\mathrm{\,or\,}\left(g(x)\right)$.

Properties of a field $F$ (Characteristic $p\neq 0$)

  1. $F^p=\{a^p\,|\,\,a\in F\}$ is a subfield of $F$.
    • Proof. Let $a,b\in F$. $(ab)^p=a^pb^p,1^p=1$ and $(a+b)^p=a^p+b^p$ since $p\left |\begin{pmatrix}p\\k\end{pmatrix}\right.$. Hence $a\mapsto a^p$ is an endomorphism.
  2. If $F$ is a finite, $F=F^p$
    • Proof. Since $F$ is finite, monomorphism $F\ni a\mapsto a^p\in F^p$ is onto.