Abel's irreducibility theorem

{{short description|Field theory result}}

In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel,{{citation|last=Abel|first=N. H.|authorlink=Niels Henrik Abel|title=Mémoire sur une classe particulière d'équations résolubles algébriquement|trans-title=Note on a particular class of algebraically solvable equations|journal=Journal für die reine und angewandte Mathematik|volume=1829|issue=4|pages=131–156|year=1829|doi=10.1515/crll.1829.4.131|s2cid=121388045|url=https://zenodo.org/record/1448814}}. asserts that if f(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of f(x). Equivalently, if f(x) shares at least one root with g(x) then f is divisible evenly by g(x), meaning that f(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.{{citation|title=100 Great Problems of Elementary Mathematics: Their History and Solution|first=Heinrich|last=Dörrie|publisher=Courier Dover Publications|year=1965|isbn=9780486613482|page=120|url=https://books.google.com/books?id=i4SJwNrYuAUC&pg=PA120}}.This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of {{harvtxt|Cox|2012}}. Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox's formulation is equivalent to Abel's. {{citation

| last = Cox | first = David A.

| doi = 10.1002/9781118218457

| edition = 2nd

| isbn = 978-1-118-07205-9

| publisher = John Wiley & Sons

| series = Pure and Applied Mathematics

| title = Galois Theory

| year = 2012}}.

Corollaries of the theorem include:

• If f(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x² − 2 is irreducible over the rational numbers and has $\backslash sqrt\{2\}$ as a root; hence there is no linear or constant polynomial over the rationals having $\backslash sqrt\{2\}$ as a root. Furthermore, there is no same-degree polynomial that shares any roots with f(x), other than constant multiples of f(x).
• If f(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.