Milestone map
Milestone map
2 milestones
Do the problem sets and write them up
4–8 weeks (8–10 hrs/week)
Take Introduction to Rings and Fields for free and do the problems from every week (ideals, quotient rings, polynomial rings, unique factorisation, field extensions), written up in LaTeX. Write every solution up in LaTeX and keep the source and the PDF in one public repo as you go. Lectures are free. NPTEL's proctored exam and certificate are optional and cost money; you never need them here.
Proof required
Link your public repo with the LaTeX source and the PDF of the problems from every week (ideals, quotient rings, polynomial rings, unique factorisation, field extensions), written up in LaTeX, and show your self-check (The course's worked problem sessions).
What gets checked
- Every proof is complete and correct.
- Your attempts come before the worked sessions (commit history).
- Examples and counterexamples are checked, not assumed.
Common mistakes
- Copying posted solutions: verifiers read the reasoning and the history.
- A proof that skips the hard step.
- Asking for review before the self-check.
Powstik Guide
A public repo with your LaTeX write-ups of the problems from every week (ideals, quotient rings, polynomial rings, unique factorisation, field extensions), written up in LaTeX.
Steps
- Make a public repo for this course.
- Do one problem set at a time; typeset it in LaTeX and commit.
- Self-check: Each week has "Problems" lectures. Try each problem before you watch it worked, then compare.
- Fix what the self-check shows, and say what you fixed.
- Ask three people who know the topic to check it.
Template
README Course: Introduction to Rings and Fields (NPTEL (IIT Madras)) What is here (problem sets, PDF): Self-check result: The hardest proof and why:
What gets sent back
- Answers with no reasoning.
- Photos of handwriting instead of typeset work.
- Self-check not shown.
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Every proof is complete and correct.
- Your attempts come before the worked sessions (commit history).
- Examples and counterexamples are checked, not assumed.
- Pick two proofs and check every step: each argument must be correct and complete, not just present.
- Ask about one step; the answer should be clear.
You'll sign in first, then come straight back here.
Build a finite field
1–3 weeks (5–8 hrs/week)
Use what you learned. Make an expository write-up that builds a field with 4, 8 or 9 elements from an irreducible polynomial, proves it is a field and gives its addition and multiplication tables. Typeset it in LaTeX (or a notebook with LaTeX maths) in your public repo.
Proof required
Link an expository write-up that builds a field with 4, 8 or 9 elements from an irreducible polynomial, proves it is a field and gives its addition and multiplication tables, in your public repo.
What gets checked
- Irreducibility is proved.
- The field axioms are checked, with the inverse of every non-zero element shown.
- The tables are right (checked by a short program).
Common mistakes
- A gap in the main argument.
- A result stated but not proved.
Powstik Guide
An expository write-up that builds a field with 4, 8 or 9 elements from an irreducible polynomial, proves it is a field and gives its addition and multiplication tables.
Steps
- Write one line: what you will prove or show.
- Draft it, then read it cold the next day.
- Ask someone to read it and mark the first step they doubt.
- Fix it and typeset the final version.
- Ask three people to check it.
Template
Title: What this shows: Main result (statement): Proof: What I would look at next:
What gets sent back
- A main step left unproved.
- Not clearly the person's own work.
- Claims more than it proves.
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Irreducibility is proved.
- The field axioms are checked, with the inverse of every non-zero element shown.
- The tables are right (checked by a short program).
- Every argument must be correct and complete; mark the first gap you find.
- Check that the work is the person's own.
You'll sign in first, then come straight back here.