Polynomials¶
Recall (doc/postdoc). Not a first course.
Job: free \(K\)-algebra \(K[x]\) (or \(K[x_1,\ldots,x_n]\)); Euclidean algorithm in one variable.
Core¶
\[
p=\sum_{k=0}^n a_k x^k\in K[x],\quad a_n\neq0\Rightarrow\deg p=n
\]
| Fact | Statement |
|---|---|
| Degree add | \(\deg(pq)=\deg p+\deg q\) (\(K\) integral domain) |
| Degree sum | \(\deg(p+q)\le\max(\deg p,\deg q)\) |
| Division | \(f=qg+r\), \(\deg r<\deg g\) (\(g\neq0\)) |
| Remainder thm | \(f(a)=\) remainder of \(f\) on \(x-a\) |
| Factor thm | \(f(a)=0\iff (x-a)\mid f\) |
| FTA (\(\mathbb{C}\)) | every nonconst splits into linears |
Not polynomials: \(1/x\), \(\sqrt{x}\), \(|x|\), \(a^x\) as formal expressions in \(K[x]\).
UFD: \(K\) field \(\Rightarrow K[x]\) Euclidean ⇒ PID ⇒ UFD. Multivariable: UFD not PID.
Used later for¶
- Factoring / Galois: roots, splitting fields
- Rational expressions: \(\operatorname{Frac}(K[x])\)
- Linear algebra: characteristic polynomial
- Algebraic geometry: \(K[x_1,\ldots,x_n]/I\)
Check¶
Is \(\deg(p+q)=\max(\deg p,\deg q)\) always?
No — leading terms may cancel; inequality is \(\le\). Equality if leading degrees differ.