One presentation, three scalar settings: an algebra manuscript
Develop one sustained manuscript about M(t)=[[t,1],[0,t]]: certify a polynomial presentation change, specialize to integer abelian groups, and interpret the field-polynomial quotient as an operator. Explain exactly which conclusions survive changing scalars. Preserve predictions, independent verification routes and a revision and defense record. This public supported project does not automatically establish mastery, independent authorship or an unseen assessment result.
Planning estimate: 16 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Before calculating, label which coordinates a right-hand change V acts on and which a left-hand change U acts on. The two copies of R² have different jobs. State the scalar ring R before interpreting the quotient.
Preparation and working agreement
Use column relations: over a commutative unital ring R, a matrix M presents R²/im M. Its columns are relations among the ambient generators e₁,e₂. Integer specialization means evaluate every polynomial at the same integer k. Over F[t], F is a declared field and multiplication by t becomes an F-linear operator on the quotient.
Return as needed to Modules and maps, Integer presentations, Smith reduction and classification, Polynomial Smith form, Operator presentations and Rational canonical form in this course. In particular Z[t] is not a PID. An explicit invertible certificate for this particular matrix does not prove a Smith theorem for all matrices over Z[t].
Submit a scope contract, dated predictions, exact matrix receipts, quotient maps with inverse checks, parameter tables, a field comparison, and a revised manuscript with a defense record. The supported noncoding route uses paper or typed exact algebra; the implementation variant adds reproducible symbolic or exact-integer calculations. These are variants of the same project, and neither numerical agreement nor code execution replaces the general proofs.
The workbench saves written text, not uploaded files or executed code. Each milestone draft is bounded at 8,000 characters. Summarize evidence and reference clearly named calculations or code artifacts retained separately; those artifacts are not automatically uploaded or verified.
The supplied solutions are example guidance, not learner evidence. Record when examples, transfer prompts or solutions were viewed, and disclose sources and assistance. A second calculation made after reading an answer is a verification, not an unseen independent prediction. Suggested hours are an unvalidated author estimate.
Keep genuine review comments and dated revisions. If no independent reviewer is available, label review pending and record a self-audit separately. Do not invent reviewer approval or a failed conjecture. Public transfer exposure remains disclosed after reset or revision.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
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0 of 6 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Freeze the question and predict the quotient
Before consulting later solutions, write a contract for the three scalar settings Z[t], Z after t=k, and F[t]. For M(t) and D(t)=diag(t,t), derive the column relations. Predict whether their quotients agree, which standard generator might generate the entire M quotient, and what could go wrong at k=0 or a unit. Preserve your first predictions, including uncertainty. Draw the relation map R²→R² and label which copy contains coefficients of relations and which contains ambient generators.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive and verify a reversible polynomial certificate
Derive U=[[1,0],[t,−1]] and V=[[0,1],[1,−t]] from reversible operations, then verify UMV=diag(1,t²) without numerical substitution. Give polynomial inverses and the induced cokernel map. Audit the tempting statement that V changes the ambient generator coordinates. Explain why evaluating this certificate at any integer remains valid, and why its existence does not make Z[t] a PID.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Classify every integer specialization and verify the maps
Classify coker M(k) and coker diag(k,k) for every integer k, explicitly separating 0, ±1 and |k|>1. Exhibit a quotient map with e₂ as cyclic generator, prove its entire kernel, and compare orders and exponents. Use determinantal divisors as a second invariant-based route. Include exact tables for k=−3,0,1,2 and one additional chosen integer; record which checks are examples and which establish the general result.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Recover an operator from the polynomial module
Now work over an arbitrary field F. Construct explicit isomorphisms for the two polynomial quotients, determine their F-dimensions, and write multiplication by t in two stated bases for the M quotient. Compute characteristic and minimal polynomials and decide similarity to the operator from D. Explain why row/column equivalence of polynomial presentations is not a constant similarity transformation of the displayed presentation matrices.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Audit two genuinely different verification routes
Freeze a draft classification before verification. Build a certificate table with two routes: reversible matrix identities, and direct generator relations with full kernel or annihilator proofs. Choose an additional integer and a field characteristic, state predictions, and check both. Deliberately audit two plausible wrong arguments: determinant equality proves isomorphic quotients, and UMV is a similarity because U and V are invertible. Explain what specialization of this explicit certificate establishes and what it does not establish for arbitrary polynomial matrices.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Revise and defend the manuscript
Produce a concise final manuscript with theorem statements, assumptions, exact certificates, all exceptional parameters, and a comparison table across scalar settings. Attach a revision log linking a genuine critique or a self-audited challenge to the affected claim and its repair. Defend orally or in a recorded written exchange: why e₂ generates, why no smaller annihilator exists, why k=0 is not a finite group, and why the two field operators are not similar. State what assistance and solution exposure occurred, which artifacts a reviewer actually inspected, and which review remains pending.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
