Patrick Dugan

Patrick Dugan

he/him

M.S.E./B.S.E. in CS and B.A. in Math at Penn. Combinatorics, learning theory, ML systems.

Fields

less than 1 minute read

Published:

This is part 3 of a three-part series, written in my junior year of high school, that works up to proving that (Z/pZ)× is cyclic: the multiplicative group of integers mod a prime has a generator. I was teaching myself group theory at the time and worked out the route on my own, so the proofs are mine rather than the textbook’s.

This part puts the pieces together: the multiplicative group of any finite field is cyclic, and its additive group is a vector space over Z/pZ.

See it

Pick a prime p and a residue g. The path walks 1 → g → g² → … around the clock until it returns to 1, which takes ord(g) steps. Generators (blue) visit every residue, and the rest trace shorter closed orbits whose lengths divide p − 1. Under each residue is its order; click any of them, or click a point on the clock.

Read the write-up (PDF)

You May Also Enjoy

Complex Image Mapping

1 minute read

Published:

In this blog post I will be talking about a piece of software I made for my final project in my Senior year highschool art class.

Dirichlet Convolutions

1 minute read

Published:

An introduction to Dirichlet convolution, written the summer after my junior year of high school. I knew the Möbius function existed but deliberately didn’t look up its definition, so I could try to reconstruct it from what it had to do: invert the constant-one function under convolution. The write-up builds up from that.

Factoring Polynomials in Arbitrary Fields

less than 1 minute read

Published:

This is part 2 of a three-part series, written in my junior year of high school, that works up to proving that (Z/pZ)× is cyclic: the multiplicative group of integers mod a prime has a generator. I was teaching myself group theory at the time and worked out the route on my own, so the proofs are mine rather than the textbook’s.

Some Results with Kernels

less than 1 minute read

Published:

This is part 1 of a three-part series, written in my junior year of high school, that works up to proving that (Z/pZ)× is cyclic: the multiplicative group of integers mod a prime has a generator. I was teaching myself group theory at the time and worked out the route on my own, so the proofs are mine rather than the textbook’s.