Editing Field norm (section)

=== Quadratic field extensions ===
One of the basic examples of norms comes from [[quadratic field]] extensions <math>\Q(\sqrt{a})/\Q</math> where <math>a</math> is a square-free integer.

Then, the multiplication map by <math>\sqrt{a}</math> on an element <math>x + y \cdot \sqrt{a}</math> is
:<math>\sqrt{a}\cdot (x + y\cdot\sqrt{a}) = y \cdot a +  x \cdot \sqrt{a}.</math>
The element <math>x + y \cdot \sqrt{a}</math> can be represented by the vector
:<math>\begin{bmatrix}x \\ y\end{bmatrix},</math>
since there is a direct sum decomposition <math>\Q(\sqrt{a}) = \Q\oplus \Q\cdot\sqrt{a}</math> as a <math>\Q</math>-vector space.

The [[matrix (mathematics)|matrix]] of <math>m_\sqrt{a}</math> is then
:<math>m_{\sqrt{a}} = \begin{bmatrix}
0 & a \\
1 & 0
\end{bmatrix}</math>
and the norm is <math>N_{\Q(\sqrt{a})/\Q}(\sqrt{a}) = -a</math>, since it is the determinant of this matrix.

==== Norm of Q(√2) ====

Consider the [[algebraic number field|number field]] <math>K=\Q(\sqrt{2})</math>.

The Galois group of <math>K</math> over <math>\Q</math> has order <math>d = 2</math> and is generated by the element which sends <math>\sqrt{2}</math> to <math>-\sqrt{2}</math>. So the norm of <math>1+\sqrt{2}</math> is:

:<math>(1+\sqrt{2})(1-\sqrt{2}) = -1.</math>

The field norm can also be obtained without the Galois group.

Fix a <math>\Q</math>-basis of <math>\Q(\sqrt{2})</math>, say:
:<math>\{1,\sqrt{2}\}</math>.

Then multiplication by the number <math>1+\sqrt{2}</math> sends
:1 to <math>1+\sqrt{2}</math> and
:<math>\sqrt{2}</math> to <math>2+\sqrt{2}</math>.

So the determinant of "multiplying by <math>1+\sqrt{2}</math>" is the determinant of the matrix which sends the vector
:<math>\begin{bmatrix}1 \\ 0\end{bmatrix}</math> (corresponding to the first basis element, i.e., 1) to <math>\begin{bmatrix}1 \\ 1\end{bmatrix}</math>,
:<math>\begin{bmatrix}0 \\ 1\end{bmatrix}</math> (corresponding to the second basis element, i.e., <math>\sqrt{2}</math>) to <math>\begin{bmatrix}2 \\ 1\end{bmatrix}</math>,
viz.:

:<math>\begin{bmatrix}1 & 2 \\1 & 1 \end{bmatrix}.</math>

The determinant of this matrix is −1.