Fields

A field $\field$ can either be $\comp$ or $\real$. A field is a set of atleast two distinct elements ${0, 1}$ satisfying all the properties laid out here along with addition and multiplication defined.

To define higher dimensional fields, we take the help of lists. For a positive integer $n$ \begin{align} \field^{n} = {(x_{1}, x_{2}, \ldots, x_{n}) : x_{j} \in \field \; \forall j = 1, 2, \ldots, n} \end{align}

which is a set of all lists satisfying the above condition. Also, if $x \in \field^{n}$, then $x_{j}$ is called the $j^{th}$ coordinate of $x$.

Similar to complex fields, addition is defined as summing up the coordinates individually \begin{align} (x_{1}, x_{2}, \ldots, x_{n}) + (y_{1}, y_{2}, \ldots, y_{n}) &= (x_{1} + y_{1}, x_{2} + y_{2}, \ldots, x_{n} + y_{n}) \end{align}

When dealing with $\field^{n}$, we will often just refer to an element in it with a letter $x$ rather than a list to make things more manageable.

$\field^{n}$ satisfies the following properties

  • Commutativity

    $x + y = y + x \quad \forall \; x, y \in \field^{n}$ \item \emph{Additive Identity Element}\newline There exists element $0$ such that\newline $x + 0 = x \quad \forall \; x \in \field^{n}$ and $0 = (0, 0, \ldots 0) \quad n $ times

  • Additive inverse \begin{gather} \forall \quad x \in \field^{n}, \exists \; -x = (-x_{1}, -x_{2}, \ldots, -x_{n}) \; \text{such that}\newline x + (-x) = 0 \end{gather}
  • Scalar Multiplication \begin{align} \lambda \times (x_{1}, x_{2}, \ldots, x_{n}) &= (\lambda x_{1}, \lambda x_{2}, \ldots, \lambda x_{n})\newline \text{where} \; (x_{1}, x_{2}, \ldots, x_{n}) &\in \field^{n}, \lambda \in \field \end{align}