Linear Algebra MT23, Fields
Flashcards
When is a set $\mathbb F$ with two binary operations $+$ and $\times$ a field?
If both $(\mathbb F, +, 0)$ and $(\mathbb F - \{0\}, \times, 1)$ are abelian groups and the distribution law holds:
\[(a + b) c = ac + bc\]What is the characteristic of a field $\mathbb F$?
The smallest integer $p$ such that
\[1 + 1 + \cdots + 1 \text{ (p times)} = 0\]and if no such $p$ exists, then the characteristic is $0$.
Alternative, equivalent definition: Consider the additive subgroup $\langle 1 \rangle$ where $1$ is the identity of $\mathbb F$. Then this is an Abelian group, and so isomorphic to $\mathbb Z / d \mathbb Z$ for some $d \ge 0$. $d$ is the characteristic.
If the characteristic $p$ of a field is non-zero, what must be true about $p$?
It is prime.
All fields are integral domains. What does this mean?
What does it mean for a field $\mathbb F$ to be algebraically closed?
Every non-constant polynomial in $\mathbb F[x]$ has a root in $\mathbb F$.