nLab fundamental theorem of algebra

Contents

Contents

Statement

Classically, the fundamental theorem of algebra states that

Many proofs of this theorem are known (see the references below); some use complex analysis (the reciprocal of a polynomial function cannot be bounded), some use algebraic topology (the degree of a map is invariant with respect to homotopy), and some use advanced calculus (polynomial functions on the complex numbers are open mappings). All of these proofs involve, at some level, the fact that the real numbers are Dedekind complete, which has as a consequence the fact that the real numbers are archimedean.

Algebraic proof via real closed fields

Despite its name, the fundamental theorem of algebra makes reference to a concept from analysis (the field of complex numbers). However, the analytic part may be reduced to a minimum: that the field of real numbers is real closed. This has been known essentially forever, and is easily proved using (for example) the intermediate value theorem.

The rest of the proof is algebraic and, unlike the other proof methods, applies to all real closed fields, which need not be archimedean. It is due to Emil Artin, and forms a basic chapter in the Artin–Schreier theory of real closed fields.

We recall that a real closed field is an ordered field such that every positive element has a square root, and every polynomial function of odd degree has a root. Note that the polynomial function x 2+1x^2 + 1 cannot have any root in a real closed field — or in fact in any ordered field, since we always have x 20x^2\ge 0 and hence x 2+11x^2+1 \ge 1.

Theorem

If FF is real closed, then K=F[1]K = F[\sqrt{-1}] is algebraically closed.

Proof

We must show that any irreducible polynomial pp of degree greater than 00 with coefficients in KK has a root in KK. Since FF has characteristic 00, it is a perfect field.

Thus the splitting field of pp is a finite Galois extension LL of FF, with Galois group GG. If G(2)G(2) is the Sylow 2-group of GG, then the fixed field? EE of G(2)G(2) is an odd degree extension of FF. Any αE\alpha \in E must then have an irreducible polynomial function qF[x]q \in F[x] of odd degree. But since FF is real closed, qq has a root in FF; by irreducibility, deg(q)=1\deg(q) = 1 and αF\alpha \in F, forcing E=FE = F and G=G(2)G = G(2). We have |G|>1{|G|} \gt 1 since the splitting field contains KK.

So GG is a 22-primary group. But for any prime number pp, a nontrivial finite pp-group has nontrivial center (see here), and is therefore solvable by an inductive argument. Therefore the extension L/FL/F arises from a tower of non-trivial quadratic extension?s

FL 1L n=LF \subseteq L_1 \subseteq \ldots \subseteq L_n = L

By the quadratic formula, the first field L 1L_1 arises by adjoining roots to FF of a polynomial function x 2+ax+bx^2 + a x + b,

a±a 24b2,\frac{-a \pm \sqrt{a^2 - 4b}}{2},

where a 24ba^2 - 4b is negative. Since FF is real closed, the positive element 4ba 24b - a^2 has a square root in FF, so that the roots displayed above belong to K=F[1]K = F[\sqrt{-1}]. So L 1=KL_1 = K. But KK has no nontrivial quadratic extensions by the lemma that follows, so in fact L 1=L n=KL_1 = L_n = K and the theorem is proved.

Lemma

Every element of K=F[1]K = F[\sqrt{-1}] has a square root in KK.

Proof

The proof is most easily apprehended by analogy with polar coordinate representations of complex numbers and half-angle formulas, where a square root of re iθr e^{i\theta} is given by r 1/2e iθ/2r^{1/2}e^{i\theta /2}. Let ii be a fixed square root of 1-1, and let a+bia + b i be an arbitrary element of KK, with a,bFa, b \in F. We must solve (x+yi) 2=a+bi(x + y i)^2 = a + b i, i.e., find x,yFx, y \in F that solve

x 2y 2=a,2xy=bx^2 - y^2 = a, \qquad 2x y = b

Since a 2+b 2a^2 + b^2 has a square root in FF, we may assume by homogeneity in x,yx, y that (a,b)(a, b) is on the unit circle: a 2+b 2=1a^2 + b^2 = 1. By interchanging xx and yy if need be, we may assume 0a10 \leq a \leq 1; replacing yy by y-y if need be, we may assume b0b \geq 0. Taking x,y0x, y \geq 0 such that

x 2=1+a2,y 2=1a2,x^2 = \frac{1+a}{2}, \qquad y^2 = \frac{1-a}{2},

we obtain a solution (since x 2y 2=ax^2 - y^2 = a and 4x 2y 2=b 24 x^2 y^2 = b^2).

Classical FTA via advanced calculus

As noted above, many proofs of the fundamental theorem are known. The following proof, ultimately rooted in the fact that polynomial mappings on \mathbb{C} are open mappings, has the advantage that it requires very little machinery. From what I (Todd Trimble) understand, it is close to the method used by Argand to give his proof (1814)1.

Let f:f\colon \mathbb{C} \to \mathbb{C} be a nonconstant polynomial mapping, and suppose ff has no zero.

  1. Let ss be the infimum of values |f(z)|{|f(z)|}; choose a sequence z 1,z 2,z 3,z_1, z_2, z_3, \ldots such that |f(z n)|s{|f(z_n)|} \to s. Since lim zf(z)=\lim_{z \to \infty} f(z) = \infty, the sequence z nz_n must be bounded; by the Bolzano-Weierstrass theorem it has a subsequence z n kz_{n_k} that converges to some point z 0z_0. Then |f(z n k)|{|f(z_{n_k})|} converges to |f(z 0)|{|f(z_0)|} by continuity, and converges to ss as well, so |f(z)|{|f(z)|} attains an absolute minimum ss at z=z 0z = z_0. By supposition, f(z 0)0f(z_0) \neq 0.

  2. The polynomial function ff may be uniquely written in the form

    f(z)=f(z 0)+g(z)(zz 0) nf(z) = f(z_0) + g(z)(z - z_0)^n

    where gg is polynomial function and g(z 0)0g(z_0) \neq 0. Put

    F(z)=f(z 0)+g(z 0)(zz 0) nF(z) = f(z_0) + g(z_0)(z - z_0)^n

    and choose δ>0\delta \gt 0 small so that

    |zz 0|=δ|g(z)g(z 0)|<|g(z 0)|.{|z - z_0|} = \delta \Rightarrow {|g(z) - g(z_0)|} \lt {|g(z_0)|}.
  3. FF maps the circle C={z:|zz 0|=δ}C = \{z : {|z - z_0|} = \delta\} onto a circle of radius r=|g(z 0)|δ nr = {|g(z_0)|}\delta^n centered at f(z 0)f(z_0). (This uses the fact that any complex number has an n thn^{th} root, which one can prove using polar coordinate representations. We omit the details.) Choose zCz' \in C so that F(z)F(z') is on the line segment between the origin and f(z 0)f(z_0) (we can always choose δ\delta so that also r<|f(z 0)|r \lt {|f(z_0)|}). Then

    |F(z)|=|f(z 0)|r{|F(z')|} = {|f(z_0)|} - r

    We also have

    |f(z)F(z)|=|g(z)g(z 0)||zz 0| n<|g(z 0)|δ n=r{|f(z') - F(z')|} = {|g(z') - g(z_0)|} {|z' - z_0|^n} \lt {|g(z_0)|} \delta^n = r

    according to how we chose δ\delta in 2. We conclude by observing the strict inequality

    |f(z)||F(z)|+|f(z)F(z)|<|f(z 0)|r+r=|f(z 0)|,{|f(z')|} \leq {|F(z')|} + {|f(z') - F(z')|} \lt {|f(z_0)|} - r + r = {|f(z_0)|},

    which contradicts the fact that |f(z)|{|f(z)|} attains an absolute minimum at z=z 0z = z_0.

In weak foundations

Many proofs rely explicitly on the double negation rule by first supposing that a polynomial function pp has no root and deriving a contradiction.

In constructive mathematics

In constructive mathematics, when we say “nonconstant function”, we mean “apart from every constant function”:

  • A function f:f:\mathbb{C} \to \mathbb{C} is nonconstant if for every constant function c:𝟙c:\mathbb{C} \to \mathbb{1} \to \mathbb{C}, where 𝟙\mathbb{1} is a singleton, f#cf \# c.

This follows in the tradition of constructive mathematics where non-X refers to apart from X: an irrational number in a Heyting field is a number that is apart from every rational number, a transcendental number is a number that is apart from every algebraic number, an invertible element in a Heyting field is an element that is apart from zero.

Thus, the fundamental theorem of algebra reads:

A fully choice-free constructive proof by Wim Ruitenberg (Ruitenberg 1991) exists for the modulated Cauchy complex numbers (which agree with the Dedekind complex numbers by weak countable choice or the limited principle of omniscience).

The algebraic proof of other fields of real numbers is problematic in many ways. First of all, the traditional definition of a real closed field bifurcates into multiple inequivalent definitions in constructive mathematics, since one could define odd-degree polynomials using either a denial inequality or a tight apartness relation. The definition using the denial inequality is not true for any field of real numbers, unless excluded middle is true. If one uses a tight apartness relation, then while the odd-degree polynomials have roots and every non-negative number has a square root, it does not follow that every polynomial function has a root in the real numbers, because there exist polynomial functions where we cannot determine whether the degree is odd or even.

What one really wants is that every nonconstant real polynomial function satisfies the exact intermediate value theorem. However, without weak countable choice, the exact intermediate value theorem is not true for all nonconstant polynomial functions in the real numbers. While nonconstant polynomial functions are Lipschitz continuous and thus uniformly continuous on any closed interval of the real numbers, they do not satisfy the exact intermediate value theorem without weak countable choice. There are alternative formulations of the exact intermediate value theorem where uniform continuity is replaced with being locally nonconstant, which is true of all nonconstant polynomial functions. However, this alternate formulation also relies on weak countable choice.

One might wish to use the inverse function theorem to prove that the square root of every non-negative number exists; however, the inverse function theorem only proves that the square root of every positive number exists, since the inverse function theorem leads to continuous inverses, which for the square function is the continuous square root, which is only defined on the positive real numbers. The metric square root, which is defined on the non-negative real numbers, is not continuous at 00.

The second problem is Lemma . This may fail in a topos (such as sheaves over \mathbb{C}), since we may not be able to find a square root of a complex number xx (or element of K[1]K[\sqrt{-1}] more generally) if we do not know whether or not xx is apart from zero, because there is no continuous square-root function unless one assumes weak countable choice.

Fred Richman (1998) has proposed that, in the absence of WCCWCC, the FTA should be interpreted as a statement about sets of roots rather than about individual roots. He constructs a complete metric space M^ n()\hat{M}_n(\mathbb{C}) which, classically, is the space of nn-element multisets of complex numbers (and constructively is the completion of that space) and proves that every complex polynomial function pp of degree nn may be associated with a point in this space in such a way that the nn elements of that point (when viewed as a multiset, if possible, and morally in any case) are the nn roots of pp.

In RCA_0

Assuming classical logic, but weak foundations, it can be shown that FTA is true in the reverse mathematics system RCA 0RCA_0 (Tanaka-Yamazaki 2005).

History

The proof was attempted many times before Gauss gave what is accepted as the first proof in his dissertation (Gauss 1799), although this was not without issues (Gauss ‘fixed’ this proof almost 50 years later, but the last gap was not filled until the 20th century).

All proofs of this fact (of which there are many) require something analytic, in the sense that ordinary algebra will not suffice: one needs to know that the real numbers (or the complex numbers) ‘have no algebraic gaps’. For instance, the rational numbers famously don’t contain the square root of 22. The cleanest proof I know, due to Artin, that isolates this analytic germ, uses the step-ladder result that the real numbers form what is called a real closed field. This is essentially saying that non-negative real numbers have square roots, and odd degree polynomial functions have roots (anyone who has plotted a cubic can appreciate this fact). Alternatively, one can characterise real closed fields as those for whom the Intermediate Value Theorem (IVT) holds for polynomial functions. Accepting this result (which does need proof), the FTA follows using pure algebra (although not of the high-school sort).

However, it is of interest, partly theoretical, partly for the sake of finding the bare minimum needed to prove the FTA, to know an elementary proof, namely one that minimises the use of analytic techniques (for instance, the IVT for polynomial functions follows from the IVT for continuous functions, but that is like killing a mosquito with a bazooka). Gauss’ second proof (Gauss 1866) is elementary (and predates Artin’s by a long time). Since Gauss lacked modern algebraic techniques, some of his proof is laborious, but (Taylor 85) gives a modern gloss. (With some amusing side notes: as Taylor puts it – ‘Gauss takes the opportunity [to] be rude to his inferior contemporaries’.) Gauss’ proof, in modern language, takes up less than a page and a half, but this presupposes familiarity with some of the theory of fields (but which is pure algebra). Artin’s proof, by comparison, drawing on major theorems can be given in half a page.

It should be noted, in the context of the last statement, that proofs of the FTA can be given, relying on analytic ‘bazooka’ theorems, that are one sentence. However, to spell out the proofs of the necessary theorems, one needs a course in analysis, of some variety, so one is merely sweeping a lot under a very small rug.

References

  • Carl Gauss, Demonstratio nova theorematis functionem algebraicam rationalem integramunius variabilis in factores reales primi vel secundi gradus resolvi posse, Dissertation, Helmstedt (1799); Werke 3, 1–30 (1866) (English transl. pdf))

  • Carl Gauss, Demonstratio nova altera theorematis omnem functionem algebraicamrationalem integram unius variabilis in factores reales primi vel secundi gradus resolviposse, Comm. Soc. Reg. Sci. Göttingen 3, 107–142 (1816); Werke 3, 33–56 (1866)

  • Michael Eisermann. An Elementary Real-Algebraic Proof via Sturm Chains. pdf

    Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree translated by Paul Taylor and B. Leak (1983) (web)

  • Paul Taylor, Gauss’ Second Proof, Eureka 45 (1985) 42-47 (pdf)

A proof that Cauchy sequences of rational numbers satisfy the fundamental theorem of algebra:

The Reverse Mathematical treatment is given in

  • Kazuyuki Tanaka and Takeshi Yamazaki, Manipulating the reals in RCA 0RCA_0 in Reverse Mathematics 2001, Lecture Notes in Logic 21 (2005)

A constructive algebraic proof of the fundamental theorem of algebra for the modulated Cauchy real numbers without choice princples such as weak countable choice

  • Wim Ruitenberg, Constructing Roots of Polynomials over the Complex Numbers, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, Vol. 84, Centre for Mathematics and Computer Science, Amsterdam, 1991, pp. 107–128. (pdf)

A full formalization in the Coq proof assistant is in

  • Herman Geuvers, Freek Wiedijk, Jan Zwanenburg, A Constructive Proof of the Fundamental Theorem of Algebra without Using the Rationals (web)

See also


  1. Despite the credit given to Gauss for his demonstration of 1799, Argand’s proof is often credited as the first one that is fully rigorous. The proof given here also uses the Bolzano-Weierstrass theorem, first proven by Bolzano in 1817, making it somewhat contemporaneous. Argand is also widely credited as the one who introduced the cutting-edge idea of viewing complex numbers and their operations geometrically, which the proof here also uses (the complex plane \mathbb{C} being also known as the Argand plane).

Last revised on June 2, 2022 at 15:51:41. See the history of this page for a list of all contributions to it.