Algebra¶
Recall (doc/postdoc). Not a first course.
Job: symbolic calculus over fields; polynomials, maps \(\mathbb{R}\to\mathbb{R}\), and solution sets as affine varieties of dim \(0\).
Map¶
| Topic | Research-facing role |
|---|---|
| Elementary algebra | free algebra / evaluation homomorphisms |
| Linear equations | affine \(1\)-dim; later row reduction |
| Systems | \(Ax=b\); elimination = Gaussian |
| Inequalities | ordered fields; cones; later LP/SDP |
| Absolute value | norms, metrics, valuations |
| Polynomials | \(K[x]\) Euclidean domain |
| Factoring | factorization in UFDs |
| Rational expressions | \(K(x)\); partial fractions |
| Quadratics | degree \(2\); discriminant; Galois of \(S_2\) |
| Quadratic functions | parabola; critical points of \(C^2\) forms |
| Linear functions | affine maps \(\mathbb{R}\to\mathbb{R}\) |
| Functions | set-maps; injectivity / inverses |
| Exp & log | Lie group \((\mathbb{R}_{>0},\times)\) |
Linear algebra lives at linear-algebra/.
Density¶
\[
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\quad
y-y_1=m(x-x_1),\quad
a^2-b^2=(a+b)(a-b),\quad
\log_b x=y\iff b^y=x
\]
Check¶
In which structure is \(K[x]\) a Euclidean domain, guaranteeing gcd and unique factorization (up to units)?
For \(K\) a field: \(K[x]\) is a Euclidean domain (hence PID and UFD).