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Real Roots of Univariate Polynomials with Real Coefficients
| Content Provider | Semantic Scholar |
|---|---|
| Author | Hewitt, Christina |
| Copyright Year | 2012 |
| Abstract | 1.1 Ordered Fields Definition 1.1. K is an ordered field if it is a commutative field with a subset P ⊂ K such that 1. 0 6∈ P 2. If a ∈ K then exactly one is true: either a ∈ P or −a ∈ P or a = 0 (trichotomy) 3. P is closed under addition and multiplication. Definition 1.2. Let K be a field. A total ordering ≤ on K is compatible with the field operations if for all a, b, c ∈ K 1. a ≤ b⇒ a+ c ≤ b+ c, 2. a ≤ b, c ≥ 0⇒ ac ≤ bc. Proposition 1.3. K is an ordered field if and only if it has a total ordering that is compatible with the field operations. Proof. If (K,P ) is an ordered field, then we can define the total ordering for a, b ∈ K a ≤ b ⇔ b− a ∈ P or b− a = 0. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://aszanto.math.ncsu.edu/MA722/ln-05.pdf |
| Alternate Webpage(s) | http://www4.ncsu.edu/~aszanto/MA722/ln-05.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
