Math 2.0

Field theory and polynomials Definition

Adjoining a root of a polynomial

Builds on K[x]/(f) is a field iff f is irreducibleIrreducible polynomialQuotient ring

Definition

Let KK be a field and let f∈K[x]f \in K[x] be irreducible. The field obtained by adjoining a root of ff to KK is the quotient ring K[x]/(f)K[x]/(f). The map K→K[x]/(f)K \to K[x]/(f), c↦c+(f)c \mapsto c + (f), identifies KK with a subfield, and the class α=x+(f)\alpha = x + (f) is a root of ff.

Remarks

The quotient is a field, because ff is irreducible. Its degree over KK is deg⁡f\deg f, with the basis 1,α,…,αdeg⁡f−11, \alpha, \dots, \alpha^{\deg f - 1}. For example, adjoining a root of x2+1x^2 + 1 to ℝ\mathbb{R} gives a field isomorphic to ℂ\mathbb{C}.

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIdeal, definition, no evidence levelIdealdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionIdeal generated by a subset, definition, no evidence levelIdeal generated by a …definitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionField, definition, no evidence levelFielddefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionPrincipal ideal, definition, no evidence levelPrincipal idealdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinitionThe only ideals of a field are zero and the whole field, statement, no evidence levelThe only ideals of a …statementIrreducible element, definition, no evidence levelIrreducible elementdefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionEuclidean domain, definition, no evidence levelEuclidean domaindefinitionCorrespondence theorem for rings, statement, no evidence levelCorrespondence theore…statementIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionPrincipal ideal domain, definition, no evidence levelPrincipal ideal domaindefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionDegree of a product, statement, no evidence levelDegree of a productstatementR/M is a field iff M is maximal, statement, no evidence levelR/M is a field iff M …statementIn a PID a nonzero element is irreducible iff it generates a maximal ideal, statement, no evidence levelIn a PID a nonzero el…statementDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementR[x] is an integral domain iff R is, statement, no evidence levelR[x] is an integral d…statementK[x] is a Euclidean domain, statement, no evidence levelK[x] is a Euclidean d…statementK[x]/(f) is a field iff f is irreducible, statement, no evidence levelK[x]/(f) is a field i…statementAdjoining a root of a polynomial, definition, no evidence levelAdjoining a root of a…definition
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. Commutative ring definition
  4. Ideal definition
  5. Polynomial ring definition
  6. Unit of a ring definition
  7. Zero divisor definition
  8. Ideal generated by a subset definition
  9. Maximal ideal definition
  10. Two-sided ideal definition
  11. Field definition
  12. Degree of a polynomial definition
  13. Integral domain definition
  14. Principal ideal definition
  15. Quotient ring definition
  16. The only ideals of a field are zero and the whole field statement
  17. Irreducible element definition
  18. Leading coefficient definition
  19. Euclidean domain definition
  20. Correspondence theorem for rings statement
  21. Irreducible polynomial definition
  22. Principal ideal domain definition
  23. Monic polynomial definition
  24. Degree of a product statement
  25. R/M is a field iff M is maximal statement
  26. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  27. Division algorithm for polynomials statement
  28. R[x] is an integral domain iff R is statement
  29. K[x] is a Euclidean domain statement
  30. K[x]/(f) is a field iff f is irreducible statement
  31. Adjoining a root of a polynomial definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Splitting_field, The field Ki[X]/(f(X))
About this page
Tag0M893X7YY7
URIm2://0M893X7YY7
Kinddefinition
WritingW0 stub
MSC12F05
DefinesAdjoining a root of a polynomial
UsesK[x]/(f) is a field iff f is irreducible, Irreducible polynomial, Quotient ring

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