Abel–Ruffini theorem

The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to polynomial equations of degree five or higher.

1 Misinterpretation
2 Lower-Degree Polynomials
3 Quintics and Higher
4 Proof
5 History
6 See also
7 References

Misinterpretation

The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. In fact, the opposite is true: every polynomial with only real or complex coefficients can be solved with a complex number; this is the fundamental theorem of algebra. Although these solutions cannot always be expressed exactly with radicals, they can be computed to any desired degree of accuracy using numerical methods such as the Newton-Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees.

The theorem only concerns the form that such a solution must take. The theorem says that not all higher-degree equations have solutions which can be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the equation's coefficients. Some polynomials of arbitrary degree are indeed solvable with a finite number of such operations. The simplest nontrivial example is the monomial equation $ax^n = b$ , whose solution is $\sqrt[n]{b \over a}.$

Lower-Degree Polynomials

The solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multiplication, division, and square roots, using the familiar quadratic formula: The roots of the following equation are shown below: $\textstyle{ax^2 + bx + c = 0}$

$x=\frac{-b \pm \sqrt {b^2-4ac\ }}{2a}.$

Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, had been known since the 16th century.

Quintics and Higher

The Abel–Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed. The equation x⁵ - x + 1 = 0 is an example. (See Bring radical.) Some other fifth degree equations can be solved by radicals, for example x⁵ - x⁴ - x + 1 = 0, which factorizes to (x-1)(x-1)(x+1)(x+i)(x-i) = 0. The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannot was given by Évariste Galois and is now part of Galois theory: a polynomial equation can be solved by radicals if and only if its Galois group is a solvable group.

Today, in the modern algebraic context, we say that second, third and fourth degree polynomial equations can always be solved by radicals because the symmetric groups S₂, S₃ and S₄ are solvable groups, whereas S_n is not solvable for n≥5.

Proof

The following proof is based on Galois theory. One of the fundamental theorems of Galois theory states that an equation is solvable in radicals if and only if it has a solvable Galois group, so the proof of the Abel-Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree.

Let $y_1$ be a real number transcendental over the field of rational numbers $Q$ , and let $y_2$ be a real number transcendental over $Q(y_1)$ , and so on to $y_5$ which is transcendental over $Q(y_1, y_2, y_3, y_4)$ . These numbers are called independent transcendental elements over Q. Let $E = Q(y_1, y_2, y_3, y_4, y_5)$ and let

$f(x) = (x - y_1)(x - y_2)(x - y_3)(x - y_4)(x - y_5) \in E[x].$

Multiplying $f(x)$ out yields the elementary symmetric functions of the $y_n$ :

$s_1 = y_1 + y_2 + y_3 + y_4 + y_5$

$s_2 = y_1y_2 + y_1y_3 + \cdots + y_4y_5$

and so on up to

$s_5 = y_1y_2y_3y_4y_5$

The coefficient of $x^n$ in $f(x)$ is thus $s_{5-n}$ . Because our independent transcendentals $y_n$ act as indeterminates over $Q$ , every permutation $\sigma$ in the symmetric group on 5 letters $S_5$ induces an automorphism $\sigma'$ on $E$ that leaves $Q$ fixed and permutes the elements $y_n$ . Since an arbitrary rearrangement of the roots of the product form still produces the same polynomial, e.g.:

$(y - y_3)(y - y_1)(y - y_2)(y - y_5)(y - y_4)$

is still the same polynomial as

$(y - y_1)(y - y_2)(y - y_3)(y - y_4)(y - y_5)$

the automorphisms $\sigma'$ also leave $E$ fixed, so they are elements of the Galois group $G(E/F)$ . Now, since $|S_5| = 5!$ it must be that $|G(E/F)| \ge 5!$ , as there could possibly be automorphisms there that are not in $S_5$ . However, since the splitting field of a quintic polynomial has at most 5! elements, $|G(E/F)| = 5!$ , and so $G(E/F)$ must be isomorphic to $S_5$ . Generalizing this argument shows that the Galois group of every general polynomial of degree $n$ is isomorphic to $S_n$ .

And what of $S_5$ ? The only composition series of $S_5$ is $S_5 \ge A_5 \ge \{e\}$ (where $A_5$ is the alternating group on five letters, also known as the icosahedral group). However, the quotient group $A_5/\{e\}$ (isomorphic to $A_5$ itself) is not an abelian group, and so $S_5$ is not solvable, so it must be that the general polynomial of the fifth degree has no solution in radicals. Since the first nontrivial normal subgroup of the symmetric group on n letters is always the alternating group on n letters, and since the alternating groups on n letters for $n \ge 5$ are always simple and non-abelian, and hence not solvable, it also says that the general polynomials of all degrees higher than the fifth also have no solution in radicals.

Note that the above construction of the Galois group for a fifth degree polynomial only applies to the general polynomial, specific polynomials of the fifth degree may have different Galois groups with quite different properties, e.g. $x^5 - 1$ has a splitting field generated by a primitive 5th root of unity, and hence its Galois group is abelian and the equation itself solvable by radicals. However, since the result is on the general polynomial, it does say that a general "quintic formula" for the roots of a quintic using only a finite combination of the arithmetic operations and radicals in terms of the coefficients is impossible. Q.E.D.

History

Around 1770, Joseph Louis Lagrange began the groundwork that unified the many different tricks that had been used up to that point to solve equations, relating them to the theory of groups of permutations. This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions for equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusive proof. The theorem, however, was first nearly proved by Paolo Ruffini in 1799, but his proof was mostly ignored and contained a gap. Ruffini assumed that a solution would necessarily be a function of the roots (in modern terms, he failed to prove that the splitting field is contained in the tower of radicals which corresponds to a solution expressed in radicals). While Cauchy felt that the assumption was minor, most historians believe that the proof was not complete until Abel proved this assumption. Ruffini's many proofs were quite innovative in using permutation groups. The theorem is generally credited to Niels Henrik Abel, who published a proof in 1824.

Insights into these issues were also gained using Galois theory pioneered by Évariste Galois. In 1885, John Stuart Glashan, George Paxton Young, and Carl Runge provided a proof using this theory.

References

Edgar Dehn. Algebraic Equations: An Introduction to the Theories of Lagrange and Galois. Columbia University Press, 1930. ISBN 0-486-43900-3.
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.
Abel's Impossibility Theorem at Everything2