Elementary Algebra¶
Recall (doc/postdoc). Not a first course.
Job: free commutative polynomial expressions; evaluation as ring homs \(R[x_1,\ldots,x_n]\to R\).
Core¶
| Object | Meaning |
|---|---|
| Variable | indeterminate / coordinate function |
| Coefficient | scalar in base ring \(R\) |
| Term / monomial | \(c\,x_1^{a_1}\cdots x_n^{a_n}\) |
| Like terms | same multi-degree |
Distributivity: \(a(b\pm c)=ab\pm ac\) — ring axiom, not a “trick.”
Equation equivalence: \(f=g\) iff \(f-g=0\). Add any \(h\); multiply by unit \(u\in R^\times\) only (else extraneous / lost solutions).
| Transform | Inverse |
|---|---|
| \(+k\) | \(-k\) |
| \(\times u\) (\(u\) unit) | \(\times u^{-1}\) |
Substitution: evaluation homomorphism \(\mathrm{ev}_a:R[x]\to R\), \(p\mapsto p(a)\); powers bind tighter than unary minus: \((-3)^2=9\), \(-3^2=-9\).
Used later for¶
- Polynomials / rings: \(R[x]\) structure
- Linear equations: solving in affine line
- Algebraic geometry: zero sets \(V(f)\)
- Logic: free vs bound, interpretation
Check¶
If you multiply both sides of an equation by \(x-1\), what solutions can appear?
Extraneous root \(x=1\) whenever both sides vanish or become equal only after that mult.; reverse only if \(x-1\) is a unit (never at \(x=1\)).