Fundamental Theorem of Algebra
Any complex, nonconstant polynomial has a zero in . Hence by induction it splits into linear factors.
We give three standard proofs. By scaling, it suffices to show this for .
Proof 1: Elementary methods (with a bit of analysis)
Main idea: If does not have a zero, take such that it attains minimum. We can move a small amount in one direction to make smaller, a contradiction.
Step 1: attains a minimum.
Note that if has absolute value at least , then is at least . This approaches infinity as approaches infinity. Hence the infimum of is the same as if we restrict it to some sufficiently large ball around 0 ( for some ). But this is a compact set (as it is closed and bounded in ), so attains a minimum here, say at . Suppose that .
Step 2: Adjusting .
For convenience, set ; that is, is a shifted and scaled version of so that is the minimum of . Write where . Then as approaches 0. However, we can choose the direction so that the second term is negative, and choosing small enough we get , contradiction.
Source: Real Analysis, Walter Rudin
Proof 2: Algebraic Topology
Main idea: Let range over the circles for different . For , is just the constant path and does not go around 0; for large , the term dominates, and the path resembles which goes around 0 times. Thus for some intermediate , passes through 0. We need to make precise what it means for to “wind” times around 0, and prove that it is invariant under “deforming” a path (without having it pass through 0). (This part seems intuitively obvious, but it is the bulk of the proof. Our proof will be longer than is necessary, so as to show an important idea in algebraic topology.)
Step 1: Smoothly deforming curves.
Suppose that has no zero. Obviously, as else is a zero.
Let . We think of this as a family of functions (or paths) . For any , we have that
(1.1) deforms continuously to ,
as is continuous in and .
Now, winds times around 0 as goes around the circle , and for sufficiently large . Thus for such a , sticks close to as varies around the circle, following around 0 times. Indeed,
(1.2) deforms continuously into .
To see this, define . Then for deforms from to in (since , none of the pass through 0).
From (1.1) and (1.2), the point deforms continuously into the function without passing through 0. A family of functions not passing though 0 connects them. It is more convenient to look at the projections of these paths on the unit circle, rotated so they have the same starting and ending point. Replacing each function by , we have that the constant path at 1 is homotopic to a path going around the circle times, i.e. one can be continuously deformed into another keeping the endpoints the same (there exists so that is continuous in ; this is just the projections of the for followed by for with suitable reparametrization). We show that this implies , i.e. is constant; then we are done. To do this, we need to classify the homotopy classes (equivalence classes under homotopy) of paths (loops) on the circle starting at ending at the same point.
Step 2: The fundamental group of the circle
The homotopy classes with the same basepoint have a group structure (which is irrelevant here, but is nevertheless an important idea) where the operation is just concatenation of paths. For simplicity represent the points on the circle with real numbers modulo 1 (i.e. that tell the number of revolutions). We show the fundamental group simply consists of the classes of paths that go around the circle times, for (the direction depending on the sign on ). Composing a path that goes around times and a path that goes around times gives a path going around paths. We prove the following.
Theorem: The fundamental group of the circle is isomorphic to , with the isomorphism , where is the path given by , i.e. it sends to the path going around the circle times.
To prove this, we find a space which maps to , such that every path in lifts to a unique path in . That is, if is the map, and are given so that , then there is a unique such that and . We want to have the property that the lift of each distinct homotopy class with is the class of a path with distinct endpoint in (so it’s easy to distinguish homotopy classes). Think of as an infinite helix above the circle, and as the projection onto the plane of the circle. has the same structure as , so we just consider the elements of as real numbers, and as mapping a number to its fractional part.
Step 2.1. Cover with the open sets and . Now the inverse image under of each is an union of disconnected open intervals in ,–simply the numbers with fractional parts in and in , respectively. Each of these open intervals is homeomorphic to its image via . is said to cover .
Step 2.2. Construct a lift locally. For each path and , must be contained in one of the open sets in (2.1), say . By continuity of , we can find a neighborhood around so that for all . Since the preimage of under is disconnected, if we fix a value for then must stay in the same component of . Since maps homeomorphically to , there is a unique and natural way to define ; just let .
Step 2.3. Construct a lift globally. The in (2.2) for an open cover for . Since this interval is compact, a finite number of them cover . There is exactly one way to define a lift with : (2.2) determines the lift on the open set containing 0; then this determines a lift on the open sets overlapping , and so on.
Step 2.3B. Do the same thing for homotopies. Each homotopy can be thought of as a map from (instead of ) to . Again we get a unique lift to . This shows that if two paths are homotopic in iff their images in are homotopic. (The forward assertion is obvious by applying .)
Step 2.4. By (2.3B), the homotopy classes in are associated with homotopy classes in starting at 0 and ending in . Such homotopies are classified by their endpoints, since it is easy to see any two paths with same starting and ending point in are homotopic (exercise). Thus the homotopy classes in are just those in the image of , and these are distinct.
Source: Algebraic Topology, Allen Hatcher
Proof 3: Complex Analysis
Main Idea: If has no zeros, then is bounded and holomorphic (complex differentiable) on the whole complex plane. (It is bounded because is either constant or approaches infinity as approaches infinity.) Then it must be constant by Liouville’s Theorem (see below).
Index: The index (or winding number) of with respect to the curve is defined by
If is a piecewise differentiable curve not passing through , then the index is an integer. The index offers another way of describing what we mean by a curve going times around 0, without reference to the fundamental group.
Cauchy’s Integral Formula: Suppose that is holomorphic in an open disk and let be a closed curve in the disk. Then for any not on ,
Liouville’s Theorem: A function which is holomorphic and bounded in the whole plane is constant.
Proof: Apply Cauchy’s Integral Formula to a circle of radius around to get
If , then this gives . By taking to infinity, we get that , i.e. is constant.
Source: Complex Analysis, Lars Ahlfors
Proof 4: Field Theory
We first show that all polynomials with real coefficients are reducible over the complex numbers, by induction on the highest power of 2 dividing the degree. For odd degree, the the statement follows since the polynomial has different signs near at . Now assuming the induction hypothesis, suppose where is odd. Choose a splitting field of , and write . Consider the polynomial
Its degree is . Since its coefficients are symmetric polynomials in the , by hypothesis it has a complex zero, i.e. is real for some . Since this is true for infinitely many values of , we must have that is real for all some . This means and are both real. Then are roots of the quadratic so they are complex roots of . This concludes the induction.
Next for an arbitrary polynomial , consider the real polynomial . (We take the conjugate of the coefficients, not .) By the above, it factors entirely into linear factors. divides , so it splits as well.