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.

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.

This part proves that every finite abelian group splits as a direct product of its p-parts, using kernels of the maps x ↦ xpk.

Part 3 ends with an interactive version you can play with: watch (Z/pZ)× turn out cyclic.

Read the write-up (PDF)

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Dirichlet Convolutions

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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.

Fields

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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.

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.