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 \(n\)

\[ \begin{aligned} \field^{n} = \{(x_{1}, x_{2}, \ldots, x_{n}) : x_{j} \in \field \; \forall j = 1, 2, \ldots, n\} \end{aligned} \]

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

\[ \begin{aligned} (x_{1}, x_{2}, \ldots, x_{n}) + (y_{1}, y_{2}, \ldots, y_{n}) &= (x_{1} + y_{1}, x_{2} + y_{2}, \ldots, x_{n} + y_{n}) \end{aligned} \]

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

\[ \begin{aligned} \lambda \times (x_{1}, x_{2}, \ldots, x_{n}) &= (\lambda x_{1}, \lambda x_{2}, \ldots, \lambda x_{n})\newline \text{where} \; (x_{1}, x_{2}, \ldots, x_{n}) &\in \field^{n}, \lambda \in \field \end{aligned} \]