Use \(A={\mathbb F}_5[u]/(u^2-2)\). Since two is not a square modulo five, this quotient is a field with 25 elements. Every element is uniquely \(a+bu\), with \(a,b\in{\mathbb F}_5\).
Reading the drawing
Rows and columns give coordinates in the quadratic quotient algebra over the field with five elements. Each cell is the exact norm residue. The displayed multiplication matrix explains how its determinant computes that norm.
Drawing labels and data 41 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
- \(N(a+bu)=a^2-2b^2\pmod5\)
| Row / column | \(b=0\) | \(b=1\) | \(b=2\) | \(b=3\) | \(b=4\) |
|---|---|---|---|---|---|
| \(a=0\) | \(0\) | \(3\) | \(2\) | \(2\) | \(3\) |
| \(a=1\) | \(1\) | \(4\) | \(3\) | \(3\) | \(4\) |
| \(a=2\) | \(4\) | \(2\) | \(1\) | \(1\) | \(2\) |
| \(a=3\) | \(4\) | \(2\) | \(1\) | \(1\) | \(2\) |
| \(a=4\) | \(1\) | \(4\) | \(3\) | \(3\) | \(4\) |
\(u^2=2\) in \(\mathbb F_{25}\)
- \(\#N^{-1}(0)=1\)
- \(\#N^{-1}(r)=6\) for \(r\ne0\)
All 25 elements are shown. Every nonzero element is a unit.
Coordinates, multiplication, and norm agree
In the basis \((1,u)\), multiplication by \(a+bu\) has matrix
The table shows every norm value. Only the zero element has norm zero. Each nonzero norm value has exactly six preimages. This makes the later character cancellation visible without estimating any sums.
The checks enumerate all 25 matrices, verify the determinant formula, and verify norm multiplicativity on every pair of field elements. They also change to a different invertible basis and check that conjugating the multiplication matrix leaves its determinant unchanged.
This model is the reduction of the quadratic integer algebra \({\mathbb Z}[\sqrt2]\) modulo five. Its finite arithmetic illustrates the determinant construction used in the continuation; it is not a computation of all ideal classes or a general ideal-counting theorem.
Read this as
A quotient norm is built from the actual multiplication operator. Its basis independence is a determinant identity, not a choice of convenient coordinates.