One presentation, three scalar settings: an algebra manuscript: changed case
Address this changed case in a fresh argument. Explain which earlier conclusions survive and which require new evidence. This is public practice with available guidance; it is not an untouched examination. Your responses and guidance exposure are stored separately from the main project.
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Transfer: two distinct roots merge after changing characteristic
This is a public changed case, not a secure test. Before opening its solution, study Mₐ,ᵦ(t)=[[t−a,1],[0,t−b]]. Derive a reversible certificate and the exact cyclic module map. Set a=1,b=3 and compare the resulting operator over Q and F₂: give a constant operator matrix, characteristic and minimal polynomials, eigenvectors or a non-diagonalizability proof, and explain why changing characteristic differs from extending a field. Optionally specialize t=k over Z and classify all integer quotients, including k=1,3. Record exposure, predictions and a defense of what survives from the primary dossier.
Structure your argument and preserve revisions
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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.
Targeted lesson returns
Transfer: two distinct roots merge after changing characteristic
