Difference between revisions of "Fundamental Theorem of Algebra"
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− | + | The '''fundamental theorem of algebra''' states that every [[nonconstant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some [[analysis]], since the result depends on certain properties of the complex numbers that are most naturally described in [[topology | topological]] terms. | |
− | The '''fundamental theorem of algebra''' states that every [[nonconstant]] [[polynomial]] with [[complex number|complex]] | ||
− | + | It follows from the [[division algorithm]] that every complex polynomial of degree <math>n</math> has <math>n</math> complex roots, counting multiplicities. In other words, every polynomial over <math>\mathbb{C}</math> splits over <math>\mathbb{C}</math>, or decomposes into linear factors. | |
− | == Proof == | + | == Proofs == |
− | + | ||
+ | === Proof by Liouville's Theorem === | ||
+ | |||
+ | We use [[Liouville's Boundedness Theorem]] of [[complex analysis]], which says that every [[bounded]] [[entire function]] is [[constant]]. | ||
+ | |||
+ | Suppose that <math>P(z)</math> is a complex polynomial of degree <math>n</math> with no complex roots; without loss of generality, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bounded when <math>n=0</math>; we now consider the case when <math>n>0</math>. | ||
+ | |||
+ | Let <math>R</math> be the sum of absolute values of the coefficients of <math>P</math>, so that <math>R \ge 1</math>. Then for <math>\lvert z \rvert \ge R</math>, | ||
+ | <cmath> \lvert P(z) \rvert \ge \lvert z^n \rvert - (R-1) \lvert z^{n-1} \rvert | ||
+ | = \lvert z^{n-1} \rvert \cdot \bigl[ \lvert z \rvert - (R-1) \bigr] | ||
+ | \ge R^{n-1} . </cmath> | ||
+ | It follows that <math>1/P(z)</math> is a bounded entire function for <math>\lvert z \rvert > R</math>. On the other hand, by the [[Heine-Borel Theorem]], the set of <math>z</math> for which <math>\lvert z \rvert \le R</math> is a [[compact set]] so its image under <math>1/P</math> is also compact; in particular, it is bounded. Therefore the function <math>1/P(z)</math> is bounded on the entire complex plane when <math>n>0</math>. | ||
+ | |||
+ | Now we apply Liouville's theorem and see that <math>1/P(z)</math> is constant, so <math>P(z)</math> is a constant polynomial. The theorem then follows. <math>\blacksquare</math> | ||
+ | |||
+ | === Algebraic Proof === | ||
+ | |||
+ | Let <math>P(x)</math> be a polynomial with complex coefficients. Since <math>F(x) = P(x) \overline{P(x)}</math> is a polynomial with real coefficients such that the roots of <math>P</math> are also roots of <math>F</math>, it suffices to show that every polynomial with ''real'' coefficients has a complex root. To this end, let the degree of <math>F</math> be <math>d = 2^n q</math>, where <math>q</math> is odd. We induct on the quantity <math>n</math>. | ||
+ | |||
+ | For <math>n=0</math>, we note that for sufficiently large negative real numbers <math>x</math>, <math>F(x) < 0</math>; for sufficiently large positive real numbers <math>x</math>, <math>F(x) > 0</math>. It follows from the [[Intermediate Value Theorem]] that <math>F(x)</math> has a real root. | ||
+ | |||
+ | Now suppose that <math>n > 0</math>. Let <math>C</math> be a [[splitting field]] of <math>F</math> over <math>\mathbb{C}</math>, and let <math>x_1, \dotsc, x_d</math> be the roots of <math>F</math> in <math>C</math>. | ||
+ | |||
+ | Let <math>c</math> be an arbitrary real number, and let <math>y_{c,i,j} = x_i + x_j + cx_ix_j</math> for <math>1 \le i \le j \le d</math>. Let | ||
+ | <cmath> G_c(x) = \prod_{1 \le i \le j \le d} (x-y_{c,i,j}) . </cmath> | ||
+ | The coefficients of <math>G</math> are symmetric in <math>x_1, \dotsc, x_d</math>. Therefore they can be expressed as linear combinations of real numbers times the [[elementary symmetric polynomial]]s in <math>x_1, \dotsc, x_n</math>; thus they are real numbers. Since the degree of <math>G_c</math> is <math>\binom{d+1}{2} = 2^{n-1}q(d+1)</math>, it follows by inductive hypothesis that <math>G_c</math> has a complex root; that is, <math>y_{c,i(c),j(c)} \in \mathbb{C}</math> for some <math>1 \le i(c) \le j(c) \le d</math>. | ||
+ | |||
+ | Now, since there are infinitely many real numbers but only finitely many integer pairs <math>(i,j)</math> with <math>1 \le i \le j \le d</math>, it follows that for two distinct numbers <math>c,c'</math>, <math>(i(c),j(c)) = (i(c'),j(c')) = (i,j)</math>. It follows that <math>x_i + x_j</math> and <math>x_ix_j</math> are both complex numbers, so <math>x_i</math> and <math>x_j</math> satisfy a quadratic equation with complex coefficients. Hence they are complex numbers. Therefore <math>F</math> has a complex root, as desired. <math>\blacksquare</math> | ||
+ | |||
+ | == References == | ||
+ | |||
+ | * Samuel, Pierre (trans. A. Silberger), ''Algebraic Theory of Numbers'', Dover 1970, ISBN 978-0-486-46666-8 . | ||
+ | * [http://www.cut-the-knot.org/fta/analytic.shtml Proofs of the Fundamental Theorem of Algebra on Cut the Knot] | ||
== See also == | == See also == | ||
* [[Algebra]] | * [[Algebra]] | ||
− | |||
− | |||
− | |||
− | [[Category:Complex | + | [[Category:Algebra]] |
+ | [{Category:Complex analysis]] | ||
+ | [[Category:Polynomials]] | ||
+ | [[Category:Theorems]] |
Latest revision as of 14:22, 11 January 2024
The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has a complex root. In fact, every known proof of this theorem involves some analysis, since the result depends on certain properties of the complex numbers that are most naturally described in topological terms.
It follows from the division algorithm that every complex polynomial of degree has
complex roots, counting multiplicities. In other words, every polynomial over
splits over
, or decomposes into linear factors.
Proofs
Proof by Liouville's Theorem
We use Liouville's Boundedness Theorem of complex analysis, which says that every bounded entire function is constant.
Suppose that is a complex polynomial of degree
with no complex roots; without loss of generality, suppose that
is monic. Then
is an entire function; we wish to show that it is bounded. It is clearly bounded when
; we now consider the case when
.
Let be the sum of absolute values of the coefficients of
, so that
. Then for
,
It follows that
is a bounded entire function for
. On the other hand, by the Heine-Borel Theorem, the set of
for which
is a compact set so its image under
is also compact; in particular, it is bounded. Therefore the function
is bounded on the entire complex plane when
.
Now we apply Liouville's theorem and see that is constant, so
is a constant polynomial. The theorem then follows.
Algebraic Proof
Let be a polynomial with complex coefficients. Since
is a polynomial with real coefficients such that the roots of
are also roots of
, it suffices to show that every polynomial with real coefficients has a complex root. To this end, let the degree of
be
, where
is odd. We induct on the quantity
.
For , we note that for sufficiently large negative real numbers
,
; for sufficiently large positive real numbers
,
. It follows from the Intermediate Value Theorem that
has a real root.
Now suppose that . Let
be a splitting field of
over
, and let
be the roots of
in
.
Let be an arbitrary real number, and let
for
. Let
The coefficients of
are symmetric in
. Therefore they can be expressed as linear combinations of real numbers times the elementary symmetric polynomials in
; thus they are real numbers. Since the degree of
is
, it follows by inductive hypothesis that
has a complex root; that is,
for some
.
Now, since there are infinitely many real numbers but only finitely many integer pairs with
, it follows that for two distinct numbers
,
. It follows that
and
are both complex numbers, so
and
satisfy a quadratic equation with complex coefficients. Hence they are complex numbers. Therefore
has a complex root, as desired.
References
- Samuel, Pierre (trans. A. Silberger), Algebraic Theory of Numbers, Dover 1970, ISBN 978-0-486-46666-8 .
- Proofs of the Fundamental Theorem of Algebra on Cut the Knot
See also
[{Category:Complex analysis]]