Question

Existenz von \(n\)-ten Wurzeln Sei \(a \in C\) und \(n \in N\). Eine komplexe Zahl z heiBt (eine) \(n\)-te Wurzel aus \(a\), wenn \(z^{\mathrm{n}}=a\) gilt. Man zeige: Ist \(a=r(\cos \varphi+i \sin \varphi) \neq 0\), dann besitat a genau \(n\) (verschiedene) \(n\)-te Wurzeln, nämlich die komplexen Zahlen$$ z_{v}=\sqrt[n]{r}\left(\cos \frac{\varphi+2 \pi \nu}{n}+i \sin \frac{\varphi+2 \pi \nu}{n}\right), \quad 0 \leq \nu \leq n-1 $$ Im Spezialfall \(a=1\) (also \(r=1, \varphi=0\) ) erhält man Satz 1.7.

Short answer

There are exactly \(n\) distinct \(n\)-th roots of complex number \(a\).

Step-by-step solution

Understand the Problem

We are given a complex number \(a\) which is represented in polar form as \(a = r(\cos \varphi + i \sin \varphi)\). We need to show that there are exactly \(n\) distinct \(n\)-th roots of \(a\), and these roots are complex numbers of the form \[z_v = \sqrt[n]{r} \left(\cos \frac{\varphi + 2 \pi u}{n} + i \sin \frac{\varphi + 2 \pi u}{n}\right), \quad 0 \leq u \leq n-1.\] This means that for each integer \(u\) between 0 and \(n-1\), there is a distinct root.

Review Polar Representation and Roots

The polar form of a complex number is given by \(r(\cos \varphi + i \sin \varphi)\). The roots are found by considering the magnitude \(r\) and the angle \(\varphi\). For an \(n\)-th root, the magnitude will be \(\sqrt[n]{r}\) and the angle of each root will incrementally differ by \(\frac{2 \pi}{n}\) to ensure their distinctness.

Determine the General Form of n-th Roots

To find the \(n\)-th roots of \(a\), we start from the basic root \(\sqrt[n]{r}\), which represents the magnitude, and then multiply by a complex unit exponentiated form \((e^{i\theta})\) where \(\theta = \frac{\varphi + 2\pi u}{n}\). Using the polar form this becomes \[\sqrt[n]{r} \left(\cos \frac{\varphi + 2\pi u}{n} + i \sin \frac{\varphi + 2\pi u}{n}\right)\].

Justify the n Distinct Roots

For each integer \(u\) from 0 to \(n-1\), the angle \(\theta\) ranges over distinct values. Specifically, adding \(2\pi\) to the original angle \(\varphi\) ensures a full rotation around the circle distributed evenly into \(n\) parts. This gives \(n\) unique positions on the complex plane, hence \(n\) distinct roots.

Special Case: a = 1

For the special case when \(a = 1\), the polar representation simplifies as \(r=1\) and \(\varphi=0\). Therefore, the roots are \(z_v = \cos \frac{2\pi u}{n} + i \sin \frac{2\pi u}{n}\), reflecting the fact that \(1\) in complex terms spreads evenly on the unit circle, consistently giving rise to the known identity that \(1\) has exactly \(n\) distinct \(n\)-th roots at regular intervals on the circle.

Key concepts

nth roots of unity

The concept of nth roots of unity is a fundamental area in complex analysis. It deals with finding all the possible solutions to the equation \(z^n = 1\). These solutions are known as the "nth roots of unity." To solve for these roots, we consider the complex number 1, which can be expressed in polar form as \(1 = e^{i0} = ext{{cos }} 0 + i ext{{sin }} 0\). This simplifies the process of finding its roots.
The roots of unity are evenly spaced on the unit circle in the complex plane. Precisely, there are \(n\) such roots, each given by the expression \(e^{2 ext{{π}}iv/n}\) where \(v = 0, 1, 2, \ldots, n-1\).
Each "v-th" root is given by:

• \(z_v = ext{{cos }} \frac{2 ext{{π}}v}{n} + i ext{{sin }} \frac{2 ext{{π}}v}{n}\)

By understanding that the angle increments by \(\frac{2 ext{{π}}}{n}\), we can see that these roots are symmetrically placed on the circle, creating a regular polygon with vertices on the unit circle.

polar form of complex numbers

The polar form is an alternative way to express complex numbers, particularly useful for multiplication, division, and finding roots. Any complex number \(a\) can be expressed in polar form as \(a = r( ext{{cos }} \varphi + i ext{{sin }} \varphi)\), where:

• \(r\) is the magnitude of \(a\), calculated as \(\sqrt{x^2 + y^2}\) where \(x\) and \(y\) are the real and imaginary components respectively.
• \(\varphi\) is the angle (argument) that \(a\) makes with the positive x-axis, found using \(\text{{tan}}^{-1}(y/x)\).

Using this form makes operations simpler:

• The magnitude \(r\) is adjusted by a root operation, such as finding the nth root.
• The angle \(\varphi\) shifts according to the rotation needed to span the full circle by starting at \(\varphi/n\) and adding \(2\text{{π}}/n\) for each subsequent root.

The beauty of the polar form lies in its ability to elegantly represent the multiplication and division of complex numbers through the addition and subtraction of their arguments respectively.

distinct roots in complex plane

In the realm of complex numbers, finding roots often results in multiple solutions. The idea of distinct roots on the complex plane becomes essential when exploring equations like \(z^n = a\). Here, each root is distinct because its angle forms a part of a symmetrical division of the plane.
The uniqueness of these roots arises from dividing a full rotation (\(360^{\circ}\) or \(2\text{{π}}\) radians) into \(n\) equal parts. Each step in this division forms a vertex in a regular n-sided polygon inscribed in the unit circle.
Thus for each integer \(u\), from 0 to \(n-1\), we can calculate the nth roots of a complex number \(a\) using:

• Magnitude: \(\sqrt[n]{r}\)
• Angle: \(\frac{\varphi + 2\text{{π}}u}{n}\)

This method ensures that all roots are evenly spaced and distinct. When transferred onto the complex plane, they align in such a way that visualizing them as vertices of a regular polygon becomes clear.
The approach highlights not only the mathematical but also the geometrical harmony in solving complex equations.