AbelRuni theorem

Not to be confused with Abels theorem.

x = bpb24ac2a

A general solution to any quadratic equation can be givenusing the quadratic formula above. Similar formulas ex-ist for polynomial equations of degree 3 and 4. But nosuch formula is possible for 5th degree polynomials; thereal solution 1.1673... to the 5th degree equation be-low cannot be written using basic arithmetic operationsand nth roots:x5 x+ 1 = 0In algebra, the AbelRuni theorem (also known asAbels impossibility theorem) states that there is nogeneral algebraic solutionthat is, solution in radicalsto polynomial equations of degree ve or higher witharbitrary coecients.[1] The theorem is named afterPaolo Runi, who made an incomplete proof in 1799,and Niels Henrik Abel, who provided a proof in 1823.variste Galois independently proved the theorem in awork that was posthumously published in 1846.[2]

1 InterpretationThe theorem does not assert that some higher-degreepolynomial equations have no solution. In fact, the op-posite is true: every non-constant polynomial equationin one unknown, with real or complex coecients, hasat least one complex number as a solution (and thus, bypolynomial division, as many complex roots as its degree,counting repeated roots); this is the fundamental theoremof algebra. These solutions can be computed to any de-sired degree of accuracy using numerical methods such asthe NewtonRaphson method or Laguerre method, andin this way they are no dierent from solutions to poly-nomial equations of the second, third, or fourth degrees.The theorem only shows that the solutions of some ofthese equations cannot be expressed via a general expres-sion in radicals.Also, the theorem does not assert that some higher-degreepolynomial equations have roots which cannot be ex-pressed in terms of radicals. While this is now known tobe true, it is a stronger claim, which was only proven a fewyears later by Galois. The theorem only shows that thereis no general solution in terms of radicals which gives theroots to a generic polynomial with arbitrary coecients.It did not by itself rule out the possibility that each polyno-mial may be solved in terms of radicals on a case-by-case

basis.

2 Lower-degree polynomialsThe solutions of any second-degree polynomial equationcan be expressed in terms of addition, subtraction, mul-tiplication, division, and square roots, using the familiarquadratic formula: The roots of the following equationare shown below:

ax2 + bx+ c = 0; a 6= 0

x =bpb2 4ac

2a:

Analogous formulas for third- and fourth-degree equa-tions, using cube roots and fourth roots, have been knownsince the 16th century.

3 Quintics and higherThe AbelRuni theorem says that there are some fth-degree equations whose solution cannot be so expressed.The equation x5 x+ 1 = 0 is an example. (See Bringradical.) Some other fth degree equations can be solvedby radicals, for example x5 x4 x + 1 = 0 , whichfactors into (x 1)(x 1)(x + 1)(x + i)(x i) = 0. The precise criterion that distinguishes between thoseequations that can be solved by radicals and those thatcannot was given by variste Galois and is now part ofGalois theory: a polynomial equation can be solved byradicals if and only if its Galois group (over the rationalnumbers, or more generally over the base eld of admit-ted constants) is a solvable group.Today, in the modern algebraic context, we say that sec-ond, third and fourth degree polynomial equations can al-ways be solved by radicals because the symmetric groupsS2, S3 and S4 are solvable groups, whereas Sn is not solv-able for n 5. This is so because for a polynomial ofdegree n with indeterminate coecients (i.e., given bysymbolic parameters), the Galois group is the full sym-metric group Sn (this is what is called the general equa-tion of the n-th degree). This remains true if the coef-cients are concrete but algebraically independent valuesover the base eld.

1

2 5 HISTORY

4 ProofThe following proof is based on Galois theory (for a shortexplanation of Arnolds proof that does not rely on priorknowledge in group theory see [3]). Historically, Runiand Abels proofs precede Galois theory and Arnolds.For a modern presentation of Abels proof see the booksof Tignol or Pesic.One of the fundamental theorems of Galois theory statesthat a polynomial f(x) F[x] is solvable by radicals overF if and only if its splitting eld K over F has a solvableGalois group,[4] so the proof of the AbelRuni theoremcomes down to computing the Galois group of the generalpolynomial of the fth degree.Let y1 be a real number transcendental over the eld ofrational numbers Q , and let y2 be a real number tran-scendental over Q(y1) , and so on to y5 which is tran-scendental over Q(y1; y2; y3; y4) . These numbers arecalled independent transcendental elements over Q. LetE = Q(y1; y2; y3; y4; y5) and let

f(x) = (xy1)(xy2)(xy3)(xy4)(xy5) 2 E[x]:

Expanding f(x) out yields the elementary symmetricfunctions of the yn :

s1 = y1 + y2 + y3 + y4 + y5

s2 = y1y2+y1y3+y1y4+y1y5+y2y3+y2y4+y2y5+y3y4+y3y5+y4y5

s3 = y1y2y3+y1y2y4+y1y2y5+y1y3y4+y1y3y5+y1y4y5+y2y3y4+y2y3y5+y2y4y5+y3y4y5

s4 = y1y2y3y4+y1y2y3y5+y1y2y4y5+y1y3y4y5+y2y3y4y5

s5 = y1y2y3y4y5:

The coecient of xn in f(x) is thus (1)5ns5n . LetF = Q(si) be the eld obtained by adjoining the sym-metric functions to the rationals (the si are all transcen-dental, because the yi are independent). Because our in-dependent transcendentals yn act as indeterminates overQ , every permutation in the symmetric group on 5letters S5 induces a distinct automorphism 0 on E thatleaves Q xed and permutes the elements yn . Since anarbitrary rearrangement of the roots of the product formstill produces the same polynomial, e.g.:

(y y3)(y y1)(y y2)(y y5)(y y4)

is still the same polynomial as

(y y1)(y y2)(y y3)(y y4)(y y5)

the automorphisms 0 also leave f xed, so they are ele-ments of the Galois group G(E/F ) . So we have shown

that S5 G(E/F ) ; however there could possibly beautomorphisms there that are not in S5 . However, sincethe relative automorphism group for the splitting eld ofa quintic polynomial has at most 5! elements, it followsthat G(E/F ) is isomorphic to S5 . Generalizing this ar-gument shows that the Galois group of every general poly-nomial of degree n is isomorphic to Sn .And what of S5 ? The only composition series of S5 isS5 A5 feg (where A5 is the alternating group onve letters, also known as the icosahedral group). How-ever, the quotient group A5/feg (isomorphic to A5 it-self) is not an abelian group, and so S5 is not solvable,so it must be that the general polynomial of the fth de-gree has no solution in radicals. Since the rst nontrivialnormal subgroup of the symmetric group on n letters isalways the alternating group on n letters, and since the al-ternating groups on n letters for n 5 are always simpleand non-abelian, and hence not solvable, it also says thatthe general polynomials of all degrees higher than the fthalso have no solution in radicals.Note that the above construction of the Galois group fora fth degree polynomial only applies to the general poly-nomial, specic polynomials of the fth degree may havedierent Galois groups with quite dierent properties,e.g. x5 1 has a splitting eld generated by a primitive5th root of unity, and hence its Galois group is abelian andthe equation itself solvable by radicals; moreover the ar-gument does not provide any rational-valued quintic thathas S5 or A5 as its Galois group. However, since the re-sult is on the general polynomial, it does say that a generalquintic formula for the roots of a quintic using only anite combination of the arithmetic operations and radi-cals in terms of the coecients is impossible. Q.E.D.

5 HistoryAround 1770, Joseph Louis Lagrange began the ground-work that unied the many dierent tricks that had beenused up to that point to solve equations, relating themto the theory of groups of permutations, in the form ofLagrange resolvents. This innovative work by Lagrangewas a precursor to Galois theory, and its failure to de-velop solutions for equations of fth and higher degreeshinted that such solutions might be impossible, but it didnot provide conclusive proof. The theorem, however,was rst nearly proved by Paolo Runi in 1799, but hisproof was mostly ignored. He had several times triedto send it to dierent mathematicians to get it acknowl-edged, amongst them, French mathematician Augustin-Louis Cauchy, but it was never acknowledged, possiblybecause the proof was spanning 500 pages. The proofalso, as was discovered later, contained an error. In mod-ern terms, Runi failed to prove that the splitting eldis one of the elds in the tower of radicals which corre-sponds to the hypothesized solution by radicals; this as-sumption fails, for example, for Cardanos solution of the

3cubic; it splits not only the original cubic but also the twoothers with the same discriminant. While Cauchy feltthat the assumption was minor, most historians believethat the proof was not complete until Abel proved thisassumption. The theorem is thus generally credited toNiels Henrik Abel, who published a proof that requiredjust six pages in 1824.[5]

Proving that some quintic (and higher) equations were un-solvable by radicals did not completely settle the matter,because the AbelRuni theorem does not provide nec-essary and sucient conditions for saying precisely whichquintic (and higher) equations are unsolvable by radicals;for example x51 = 0 is solvable. Abel was working ona complete characterization when he died in 1829.[6] Fur-thermore, Ian Stewart notes that for all that Abels meth-ods could prove, every particular quintic equation mightbe soluble, with a special formula for each equation.[7]

In 1830 Galois (at the age of 18) submitted to the ParisAcademy of Sciences a memoir on his theory of solv-ability by radicals; Galois paper was ultimately rejectedin 1831 as being too sketchy and for giving a condi-tion in terms of the roots of the equation instead of itscoecients. Galois then died in 1832 and his paper"Memoire sur les conditions de resolubilite des equationspar radicauxremained unpublished until 1846 when itwas published by Joseph Liouville accompanied by someof his own explanations.[6] Prior to this publication, Liou-ville announced Galois result to the Academy in a speechhe gave on 4 July 1843.[8] According to Allan Clark, Ga-loiss characterization dramatically supersedes the workof Abel and Runi.[9]

In 1963, Vladimir Arnold discovered a topological proofof the AbelRuni theorem,[10] which served as a start-ing point for topological Galois theory.[11]

6 See also Theory of equations Constructible number

7 Notes[1] Jacobson (2009), p. 211.

[2] Galois, variste (1846). OEuvres mathmatiquesd'variste Galois.. Journal des mathmatiques pures etappliques XI: 381444. Retrieved 2009-02-04.

[3] Short proof of Abels theorem that 5th degree polynomialequations cannot be solved.

[4] Fraleigh (1994, p. 401)

[5] du Sautoy, Marcus. January: Impossibilities. Symme-try: A Journey into the Patterns of Nature. ISBN 978-0-06-078941-1.

[6] Jean-Pierre Tignol (2001). Galois Theory of AlgebraicEquations. World Scientic. pp. 232233 and 302. ISBN978-981-02-4541-2.

[7] Stewart, 3rd ed., p. xxiii

[8] Stewart, 3rd ed., p. xxiii

[9] Allan Clark (1984) [1971]. Elements of Abstract Algebra.Courier Corporation. p. 131. ISBN 978-0-486-14035-3.

[10] Tribute to Vladimir Arnold (PDF).Notices of the Amer-ican Mathematical Society 59 (3): 393. March 2012.doi:10.1090/noti810.

[11] Vladimir Igorevich Arnold. 2010.

8 References Edgar Dehn. Algebraic Equations: An Introduction

to the Theories of Lagrange and Galois. ColumbiaUniversity Press, 1930. ISBN 0-486-43900-3.

Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.),Dover, ISBN 978-0-486-47189-1

John B. Fraleigh. A First Course in Abstract Algebra.Fifth Edition. Addison-Wesley, 1994. ISBN 0-201-59291-6.

Ian Stewart. Galois Theory. Chapman and Hall,1973. ISBN 0-412-10800-3.

Abels Impossibility Theorem at Everything2

9 Further reading Peter Pesic (2003). Abels Proof: An Essay on the

Sources andMeaning ofMathematical Unsolvability.MIT Press. ISBN 978-0-262-66182-9.

10 External links Mmoire sur les quations algbriques, ou l'on d-montre l'impossibilit de la rsolution de l'quationgnrale du cinquime degr PDF - the rst proof inFrench (1824)

Dmonstration de l'impossibilit de la rsolution al-gbrique des quations gnrales qui passent le qua-trime degr PDF - the second proof in French(1826)

A video presentation on Arnolds proof.

4 11 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

11 Text and image sources, contributors, and licenses11.1 Text

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