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Mathematics LibreTexts

5.2: Other Fields

Above, we defined vector spaces over the real numbers. One can actually define vector spaces over any field. This is referred to as choosing a different base field. A field is a collection of "numbers'' satisfying properties which are listed in appendix B. An example of a field is the complex numbers, 

\[\mathbb{C}= \left\{x+iy \mid i^{2}=-1, x,y\in \Re \right\}.\]

Example 61

In quantum physics, vector spaces over \(\mathbb{C}\) describe all possible states a physical system of particles can have.
For example,

\[V= \left\{ \begin{pmatrix}\lambda \\ \mu\end{pmatrix} \mid \lambda, \mu \in \mathbb{C}\right\}\]

is the set of possible states for an electron's spin. The vectors \(\begin{pmatrix}1 \\ 0\end{pmatrix}\) and \(\begin{pmatrix}0 \\ 1\end{pmatrix}\) describe, respectively, an electron with spin "up'' and "down'' along a given direction. Other vectors, like \(\begin{pmatrix}i \\ -i\end{pmatrix}\) are permissible, since the base field is the complex numbers. Such states represent a mixture of spin up and spin down for the given direction (a rather counterintuitive yet experimentally verifiable concept), but a given spin in some other direction.

Complex numbers are very useful because of a special property that they enjoy: every polynomial over the complex numbers factors into a product of linear polynomials. For example, the polynomial $$x^{2}+1$$ doesn't factor over real numbers, but over complex numbers it factors into $$(x+i)(x-i)\, .$$ In other words, there are \(\textit{two}\) solutions to $$x^{2}=-1,$$
\(x=i\) and \(x=-i\). This property ends has far-reaching consequences: often in mathematics problems that are very difficult using only real numbers, become relatively simple when working over the complex numbers. This phenomenon occurs when diagonalizing matrices, see chapter 13.

The rational numbers \(\mathbb{Q}\) are also a field. This field is important in computer algebra: a real number given by an infinite string of numbers after the decimal point can't be stored by a computer. So instead rational approximations are used. Since the rationals are a field, the mathematics of vector spaces still apply to this special case.

Another very useful field is bits 
$$
B_2=\mathbb{Z}_2=\{0,1\}\, ,
$$
with the addition and multiplication rules
$$
\begin{array}{c|cc}
+&0&1\\\hline
0&0&1\\
1&1&0
\end{array}\qquad
\begin{array}{c|cc}
\times&0&1\\\hline
0&0&0\\
1&0&1
\end{array}
$$

These rules can be summarized by the relation \(2=0\). For bits, it follows that \(-1=1\)!

The theory of fields is typically covered in a class on abstract algebra or Galois theory.

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