Math 2.0

Field theory and polynomials Polynomials in one variable Theorem

Vieta's formulas

Builds on Elementary symmetric polynomialsSplit polynomial

Statement

Let RR be a commutative ring and let f=anxn+an−1xn−1+⋯+a0∈R[x]f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in R[x]. Suppose that f=an(x−r1)(x−r2)⋯(x−rn)f = a_n (x - r_1)(x - r_2) \cdots (x - r_n) with r1,…,rn∈Rr_1, \dots, r_n \in R. Then for 1≤k≤n1 \le k \le n, an ek(r1,…,rn)=(−1)kan−k,a_n \, e_k(r_1, \dots, r_n) = (-1)^k a_{n-k}, where eke_k is the kk-th elementary symmetric polynomial.

Remarks

If ana_n is a unit, this says ek(r1,…,rn)=(−1)kan−k/ane_k(r_1, \dots, r_n) = (-1)^k a_{n-k}/a_n. For x2+bx+c=(x−r1)(x−r2)x^2 + b x + c = (x - r_1)(x - r_2), it gives r1+r2=−br_1 + r_2 = -b and r1r2=cr_1 r_2 = c. The proof expands the product and compares coefficients. Over a field, the hypothesis says that ff splits with roots r1,…,rnr_1, \dots, r_n, counted with multiplicity.

History

The formulas are named after the French mathematician François Viète, who derived them for positive roots.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomials in several variables, definition, no evidence levelPolynomials in severa…definitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionSymmetric polynomial, definition, no evidence levelSymmetric polynomialdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionElementary symmetric polynomials, definition, no evidence levelElementary symmetric …definitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionVieta's formulas, statement, no evidence levelVieta's formulasstatement
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.
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Polynomials in several variables definition
  6. Polynomial evaluation definition
  7. Symmetric polynomial definition
  8. Root of a polynomial definition
  9. Elementary symmetric polynomials definition
  10. Split polynomial definition
  11. Vieta's formulas statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Vieta%27s_formulas, Basic formulas
  2. Web https://en.wikipedia.org/wiki/Vieta%27s_formulas, Generalization to rings
About this page
Tag0M8EWQJ1YV
URIm2://0M8EWQJ1YV
Kindstatement, theorem
WritingW1 stated
MSC12D10
StatusThis copy of the site has no build data. Run m2 status 0M8EWQJ1YV.
UsesElementary symmetric polynomials, Split polynomial
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