Math 2.0

Commutative algebra Domains and factorization Theorem

Gauss's lemma

Builds on Product of primitive polynomials is primitiveUnique factorisation domainField of fractionsIrreducible polynomialPrimitive polynomial

Statement

Let RR be a unique factorisation domain with field of fractions KK and let f∈R[x]f \in R[x] be a primitive polynomial. Then ff is irreducible in R[x]R[x] if and only if ff is irreducible in K[x]K[x].

Remarks

The key step: if f=ghf = g h in K[x]K[x], then one can scale gg and hh by elements of KK so that they are in R[x]R[x] and f=ghf = g h still holds. This uses that the product of primitive polynomials is primitive. For R=ℤR = \mathbb{Z}: a primitive integer polynomial that factors over ℚ\mathbb{Q} already factors over ℤ\mathbb{Z}, into factors of the same degrees. The hypothesis that ff is primitive is necessary: 2x2x is irreducible in ℚ[x]\mathbb{Q}[x], but 2x=2⋅x2x = 2 \cdot x is a product of two nonunits in ℤ[x]\mathbb{Z}[x].

Reading path

To understand this page from the ground up, read these 20 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 ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDivisibility in a commutative ring, definition, no evidence levelDivisibility in a com…definitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionPrime element, definition, no evidence levelPrime elementdefinitionField of fractions, definition, no evidence levelField of fractionsdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionGreatest common divisor in an integral domain, definition, no evidence levelGreatest common divis…definitionUnique factorisation domain, definition, no evidence levelUnique factorisation …definitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionContent of a polynomial, definition, no evidence levelContent of a polynomi…definitionPrimitive polynomial, definition, no evidence levelPrimitive polynomialdefinitionProduct of primitive polynomials is primitive, statement, no evidence levelProduct of primitive …statementGauss's lemma, statement, no evidence levelGauss's lemmastatement
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. Polynomial ring definition
  5. Unit of a ring definition
  6. Zero divisor definition
  7. Divisibility in a commutative ring definition
  8. Field definition
  9. Integral domain definition
  10. Associate elements definition
  11. Prime element definition
  12. Field of fractions definition
  13. Irreducible element definition
  14. Greatest common divisor in an integral domain definition
  15. Unique factorisation domain definition
  16. Irreducible polynomial definition
  17. Content of a polynomial definition
  18. Primitive polynomial definition
  19. Product of primitive polynomials is primitive statement
  20. Gauss's lemma statement

Where this leads

13 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomials), Statements for unique factorization domains
About this page
Tag0QFV5Y1Z5X
URIm2://0QFV5Y1Z5X
Kindstatement, theorem
WritingW1 stated
MSC13F20
StatusThis copy of the site has no build data. Run m2 status 0QFV5Y1Z5X.
UsesProduct of primitive polynomials is primitive, Unique factorisation domain, Field of fractions, Irreducible polynomial, Primitive polynomial
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (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