finite-field
finite-field is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "log"
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- there is a field with 4 elements only if 4 is a prime power ×7
- the size of the field is a whole number between 2 and 9 ×3
- a ring of composite size has a pair of non-zero elements multiplying to zero ×1
- addition and multiplication are commutative ×1
- an irreducible polynomial of the right degree was found ×1
- and it is exactly the composite ones that leave an element without a reciprocal ×1
- every non-zero element has a reciprocal ×1
- multiplication distributes over addition ×1
- multiplication is associative ×1
- no two non-zero elements multiply to zero ×1
- the exponent of a product is the sum of the exponents ×1
- the field is closed ×1
- the number being factorised is a whole number between 1 and 1000000 ×1
- the number of primitive elements is φ(q−1) ×1
- the powers of a primitive element close back onto one ×1
- the size of the ring drawn beside it is a whole number between 0 and 12 ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A polynomial through the gaps
Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.
ComputationEvery element is a power of one of them
Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.
ComputationSeven points, seven lines
A geometry with seven points, in which every two points lie on exactly one line and every two lines meet in exactly one point. There are no parallels, the whole thing is built out of the two-element field, and one of its lines has to be drawn as a circle.
ComputationThe field with four elements
The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.