Math 2.0

Commutative algebra Definition

Root of a polynomial

Builds on Polynomial evaluation

Definition

Let RR be a ring and let f∈R[x]f \in R[x]. An element c∈Rc \in R is a root of ff if f(c)=0f(c) = 0, where f(c)f(c) is the value of ff at cc.

Remarks

The roots of x2−1x^2 - 1 in ℤ\mathbb{Z} are 11 and −1-1. The polynomial x2+1x^2 + 1 has no root in ℝ\mathbb{R}. It has the roots ii and −i-i in ℂ\mathbb{C}. By the factor theorem, over a commutative ring, cc is a root of ff if and only if x−cx - c divides ff.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Polynomial ring definition
  4. Polynomial evaluation definition
  5. Root of a polynomial definition

Where this leads

122 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Zero_of_a_function, Polynomial roots
About this page
Tag00EHX6FWC0
URIm2://00EHX6FWC0
Kinddefinition
WritingW0 stub
MSC13F20
DefinesRoot of a polynomial
UsesPolynomial evaluation

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by