In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Abel's irreducibility theorem, a
field theory result described in 1829 by
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
, asserts that if ''f''(''x'') is a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''F'' that shares a
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
with a polynomial ''g''(''x'') that is
irreducible
In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole.
Emergence plays a central role ...
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
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s in ''F''.
[.][This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of . 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. .]
Corollaries
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
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, ''x''
2 − 2 is irreducible over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s and has
as a root; hence there is no linear or constant polynomial over the rationals having
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 polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
s, then they share no roots.
References
External links
* Larry Freeman
Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility September 4, 2008.
*
Field (mathematics)
Theorems in algebra
{{Abstract-algebra-stub