In mathematics, 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 ''ƒ''(''x'') is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

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 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 below the sur ...

with a polynomial ''g''(''x'') that is irreducible over ''F'', then every root of ''g''(''x'') is a root of ''ƒ''(''x''). Equivalently, if ''ƒ''(''x'') shares at least one root with ''g''(''x'') then ''ƒ'' is divisible evenly by ''g''(''x''), meaning that ''ƒ''(''x'') can be factored as ''g''(''x'')''h''(''x'') with ''h''(''x'') also having

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

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 ''ƒ''(''x'') is irreducible, there is no lower-

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

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 (e.g. ). The set of all ra ...

s and has

\sqrt

as a root; hence there is no linear or constant polynomial over the rationals having

\sqrt

as a root. Furthermore, there is no same-degree polynomial that shares any roots with ''ƒ''(''x''), other than constant multiples of ''ƒ''(''x''). * If ''ƒ''(''x'') ≠ ''g''(''x'') are two different irreducible

monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\ ...

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) {{Abstract-algebra-stub