Roots appear in quanta
Roots appear in quanta.
American Mathematical Monthly 111 (2004), 61-63.
When you construct a field extension by adjoining one root of an irreducible polynomial, how many roots of that polynomial appear in the constructed field? That number is called the root quantum number. It places restrictions on how many roots appear in arbitrary field extensions, and it places restrictions on how a polynomial can factor over a field extension. Connections with Galois theory are given, and exercises at the end give a complete classification of the possible root quantum numbers for polynomials with rational coefficients.
2000 Mathematics Subject Classification
(MSC):
- 12F05 Field theory and polynomials | Field extensions | Algebraic extensions
- 12F10 Field theory and polynomials | Field extensions | Separable extensions, Galois theory
- 12F12 Field theory and polynomials | Field extensions | Inverse Galois theory
Key words and phrases:
- irreducible polynomial
- root quantum number
- stem field
- counting roots
- number of roots
Downloads:
Roots appear in quanta.
Last change: 08 October 2003.
My agreement with the MAA gives me permission to post the article here. For the curious: the Monthly's version (PDF) of the article (the text is identical).
Roots appear in quanta: exercise solutions.
Last change: 08 October 2003.
If you discover alternate solutions, especially shorter or more elementary ones, please let me know.
Last change: 08 October 2003.
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My agreement with the MAA gives me permission to post the article here. For the curious: the Monthly's version (PDF) of the article (the text is identical).
Roots appear in quanta: exercise solutions.
Last change: 08 October 2003.
20K
142K
126K
If you discover alternate solutions, especially shorter or more elementary ones, please let me know.
Related work:
- My exercises showed: for K=Q, each (K,n,r) except (K,2,1) occurs. This result has been generalized to K any Hilbertian field by Elizabeth Townsend.