Introduction
In the previous chapter, we stood at a crossroads. Having established that we can always expand a field to find the roots of a polynomial,
we must now ask: how do we measure the "size" or the "degrees of freedom" of this new, extended universe?
We now take the path toward continuous symmetry and geometry. Before we can understand the smooth, infinite dimensions of Lie groups and manifolds
used in modern AI, we must first understand dimension in its purest, discrete algebraic form. When we adjoin a new element to a field, we are
effectively adding a new "dimension" to a vector space.
Connection to Geometric Deep Learning (GDL)
In robotics and 3D computer vision, we constantly track degrees of freedom (for instance, a rigid body in \(SE(3)\) has 6 degrees of freedom).
In abstract algebra, the "degree of an extension" tracks the exact dimensionality of a new field over its base. The rules governing how these discrete
dimensions stack and multiply - specifically the Tower Rule - are the rigorous mathematical ancestors of dimension counting in the smooth
manifolds and Lie groups we will study next.
Characterization of Extensions
When we expand a base field \(F\) by adding a new element \(a\) to create an extension \(E\), the structure of this new universe
depends entirely on the nature of \(a\). Specifically, we must ask if \(a\) is bound by the algebraic rules of \(F\), or if it is
entirely independent.
Definition: Algebraic & Transcendental Extensions
Let \(E\) be an extension field of a field \(F\). Then \(a \in E\) is called:
-
algebraic over \(F\) if \(a\) is the zero of some nonzero polynomial in \(F[x]\).
-
transcendental over \(F\) if \(a\) is not algebraic over \(F\).
An extension \(E\) of \(F\) is called an algebraic extension of \(F\) if every element of
\(E\) is algebraic over \(F\). If \(E\) is not an algebraic extension of \(F\), it is called a
transcendental extension of \(F\). Also, an extension of \(F\) of the form \(F(a)\) is called
a simple extension of \(F\).
If an element is algebraic (like \(\sqrt{2}\) over \(\mathbb{Q}\)), it brings a finite, measurable amount of new information
to the field. If it is transcendental (like \(\pi\) or \(e\)), it brings an infinite number of linear independent powers,
blowing the dimension of the field up to infinity. Because we are building toward the study of finite-dimensional continuous
groups, we care deeply about the algebraic case.
Theorem: Characterization of Extensions
Let \(E\) be an extension of the field \(F\) and let \(a \in E\). If \(a\) is transcendental over \(F\), then
\(F(a) \cong F(x)\).
If \(a\) is algebraic over \(F\), then \(F(a) \cong F[x]/\langle p(x) \rangle\), where \(p(x)\) is the
unique monic irreducible polynomial in \(F[x]\) of minimal degree such that \(p(a) = 0\).
As established in the previous chapter, this isomorphism gives us a concrete data structure. But more importantly,
it implies a strict limit on the size of the new field. This brings us to the most crucial property of algebraic extensions:
their vector space dimension.
Finite Extensions & The Tower Rule
Here, abstract algebra beautifully intersects with Linear Algebra. Any extension field \(E\) can simply be viewed as a
vector space over its base field \(F\). For example, the complex numbers \(\mathbb{C}\) form a 2-dimensional
vector space over the real numbers \(\mathbb{R}\), with the basis \(\{1, i\}\).
Definition: Degree of an Extension
Let \(E\) be an extension field of a field \(F\). If \(E\) has dimension \(n\) as a vector space over \(F\),
\(E\) has degree \(n\) over \(F\) and write
\[
[E:F] = n.
\]
If \([E:F]\) is finite, \(E\) is called a finite extension of \(F\), and if not, \(E\) is called
an infinite extension of \(F\).
A powerful realization is that restricting the vector space dimension strictly forces all elements inside the field to be algebraic.
There is simply not enough "room" in a finite-dimensional space for an element to be transcendental.
Theorem: Finite Implies Algebraic
If \(E\) is a finite extension of the field \(F\), then \(E\) is an algebraic extension of \(F\).
Note that the converse of this theorem is not true.
Imagine, if the vector space dimension is \(n\), then any sequence of \(n+1\) elements must be linearly dependent.
For any element \(a \in E\), the \(n+1\) powers \(\{1, a, a^2, \dots, a^n\}\) must have some linear dependence relation
\(c_0 + c_1a + \dots + c_na^n = 0\). This equation is exactly a polynomial over \(F\) that \(a\) satisfies, proving \(a\) is algebraic.
Proof:
Suppose that \(E\) is a finite extension of \(F\) with dimension \([E:F] = n\), and \(a \in E\).
Consider the set of \(n+1\) elements: \(\{1, a, \ldots, a^n\}\). Since \(E\) is an \(n\)-dimensional
vector space over \(F\), any set of more than \(n\) elements must be linearly dependent.
Thus, there exist elements \(c_0, c_1, \ldots, c_n \in F\) (not all zero) such that
\[
c_n a^n + c_{n-1} a^{n-1} + \cdots + c_1 a + c_0 = 0.
\]
Clearly, then, \(a\) is a root of the nonzero polynomial
\[
f(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0.
\]
Therefore, by definition, \(a\) is algebraic over \(F\). Since this applies to any \(a \in E\), the
entire extension \(E\) is algebraic.
When building complex architectures - such as stacking layers in a deep neural network or composing transformations in a robotic arm -
we need to know how the total dimensionality behaves. In field theory, we stack extensions. The Tower Rule tells us exactly
how these discrete dimensions multiply.
Theorem: Tower Rule
If \(K\) is a finite extension of the field \(E\), and \(E\) is a finite extension of the field \(F\), then
\(K\) is a finite extension of \(F\), and their degrees multiply:
\[
[K:F] = [K:E][E:F].
\]
Proof:
Let \(X = \{x_1, x_2, \ldots, x_n\}\) be a basis for \(K\) over \(E\), and
Let \(Y = \{y_1, y_2, \ldots, y_m\}\) be a basis for \(E\) over \(F\).
We claim that the set
\[
YX = \{y_j x_i \mid 1 \leq j \leq m, \, 1 \leq i \leq n\}
\]
forms a basis for \(K\) over \(F\). To prove this, we must show that \(YX\) spans \(K\) over \(F\), and that
\(YX\) is linearly independent over \(F\).
First, let \(a \in K\). Then there are elements \(b_1, b_2, \ldots, b_n \in E\) such that
\[
a = b_1 x_1 + b_2 x_2 + \cdots + b_n x_n.
\]
and for each \(i = 1, 2, \ldots, n\), there are elements \(c_{i1}, c_{i2}, \ldots, c_{im} \in F\)
such that
\[
b_i = c_{i1} y_{1} + c_{i2} y_2 + \cdots + c_{im} y_{m}.
\]
Thus,
\[
\begin{align*}
a &= \sum_{i=1}^n b_i x_i \\\\
&= \sum_{i=1}^n \left(\sum_{j=1}^m c_{ij} y_j \right) x_i \\\\
&= \sum_{i, j} c_{ij} (y_j x_i).
\end{align*}
\]
Thus, \(YX\) spans \(K\) over \(F\).
Next, suppose there are elements \(c_{ij} \in F\) such that
\[
\begin{align*}
0 &= \sum_{i, j} c_{ij} (y_j x_i) \\\\
&= \sum_i \left(\sum_j (c_{ij} y_j)\right) x_i.
\end{align*}
\]
Then, since each \(\sum_j c_{ij} y_j \in E\) and \(X\) is a basis for \(K\) over \(E\), for each \(i\), we have
\[
\sum_j c_{ij} y_j = 0.
\]
However, each \(c_{ij} \in F\) and \(Y\) is a basis for \(E\) over \(F\), so each \(c_{ij} = 0\). Therefore,
the set \(YX\) is linearly independent over \(F\).
Since \(YX\) is a basis for \(K\) over \(F\) containing exactly \(mn\) distinct elements, we conclude that
\[
[K:F] = mn = [K:E][E:F].
\]
If we extend a field with multiple algebraic elements, say \(F(a, b)\), do we really need to track both independent dimensions?
In many cases, no. The Primitive Element Theorem acts as a form of algebraic dimensionality reduction, proving
that we can often "compress" multiple extensions into a single generated extension.
Theorem: Primitive Element Theorem
If \(F\) is a field of characteristic \(0\), and \(a\) and \(b\) are algebraic over \(F\), then
there exists a single element \(c \in F(a, b)\) such that \(F(a, b) = F(c)\).
Example: Generator Compression and Data Packing
To see how the Primitive Element Theorem works as "algebraic data compression," let's look at a concrete numerical example.
Consider the field \(E = \mathbb{Q}(\sqrt{2}, \sqrt{3})\), which is formed by adjoining both \(\sqrt{2}\) and \(\sqrt{3}\) to the
rational numbers \(\mathbb{Q}\).
-
Multidimensional Structure:
According to the Tower Rule, the degree of this extension is
\([\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = 4\). The basis is \(\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}\), meaning
it behaves like a 4-dimensional vector space over \(\mathbb{Q}\).
-
Compression via a "Primitive Element":
Surprisingly, we do not need to keep track of two separate generators. We can define a single, combined element
\(c = \sqrt{2} + \sqrt{3}\). This single element is sufficient to generate the entire field:
\[
\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3}).
\]
-
Decoding the Information:
We can see this as a "lossless" encoding by showing that the original generators can be recovered entirely through polynomial operations
on \(c\). Let's calculate the cube of \(c\):
\[
\begin{align*}
c^3 &= (\sqrt{2} + \sqrt{3})^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2\sqrt{3} + 3\sqrt{2}(\sqrt{3})^2 + (\sqrt{3})^3 \\\\
&= 2\sqrt{2} + 6\sqrt{3} + 9\sqrt{2} + 3\sqrt{3} \\\\\
&= 11\sqrt{2} + 9\sqrt{3}.
\end{align*}
\]
Now, if we subtract \(9c\) from \(c^3\), the \(\sqrt{3}\) terms beautifully cancel out:
\[
c^3 - 9c = (11\sqrt{2} + 9\sqrt{3}) - 9(\sqrt{2} + \sqrt{3}) = 2\sqrt{2}.
\]
Therefore, we can perfectly decode \(\sqrt{2}\) using the formula \(\sqrt{2} = \frac{1}{2}(c^3 - 9c)\).
Once we have \(\sqrt{2}\), we easily get \(\sqrt{3} = c - \sqrt{2}\).
The CS Perspective:
This is perfectly analogous to lossless data packing or hashing. Instead of maintaining two separate
variables (or allocating two separate memory addresses for \(\sqrt{2}\) and \(\sqrt{3}\)), we can apply a specific
weighting to sum them into a single variable (\(c\)) without any loss of structural information. The Primitive Element
Theorem guarantees that such a lossless flattening is always possible for finite extensions (in characteristic 0).
Properties of Algebraic Extensions
In algorithm design and mathematical modeling, closure is a critical property. It ensures that an operation
applied to valid inputs always produces a valid output. Before we can transition to studying smooth manifolds and continuous
groups, we must guarantee that our algebraic building blocks are perfectly stable.
Theorem: Subfield of Algebraic Elements
Let \(E\) be an extension field of the field of \(F\). Then the set of all elements in \(E\) that are
algebraic over \(F\) is a subfield of \(E\).
This means that if you take two algebraic numbers and add, subtract, multiply, or divide them, the result is
guaranteed to also be an algebraic number. Finally, this closure applies not just to elements within a single extension, but
to the extensions themselves.
Proof:
Suppose that \(a, b \in E\) are algebraic over \(F\) and \(b \neq 0\). To show they form a subfield, we must show that all
\(a+b\), \(a-b\), \(ab\) and \(a/b\) are algebraic over \(F\). To do this, it suffices to show that the extension degree
\([F(a, b):F]\) is finite, because each of these four elements is contained within \(F(a, b)\).
Since \(b\) is algebraic over \(F\), the simple extension \(F(b)\) has a finite degree over \(F\). Also, since \(a\) is
algebraic over \(F\), it is certainly algebraic over \(F(b)\). Thus, the extension \(F(a,b)\) over \(F(b)\) is also finite.
Here, by the Tower Rule, we know that
\[
[F(a, b):F] = [F(a, b):F(b)][F(b):F].
\]
Since both terms on the right are finite, \([F(a,b) : F]\) is finite. Because Finite Implies Algebraic,
every element inside \(F(a,b)\) must be algebraic over \(F\).
Since field operations are closed, \(a+b\), \(a-b\), \(ab\), and \(a/b\) are all in \(F(a,b)\). Therefore, they are all
guaranteed to be algebraic over \(F\). The algebraic elements form a subfield of \(E\).
Theorem: Algebraic over Algebraic is Algebraic
If \(K\) is an algebraic extension of \(E\), and \(E\) is an algebraic extension of \(F\), then
\(K\) is an algebraic extension of \(F\).
These theorems prove that the "algebraic universe" is closed. We can stack finite extensions on top of each other safely without
accidentally falling into infinite-dimensional (transcendental) chaos. With this secure, discrete foundation of dimensions and
extensions established, we are now ready to transition to continuous geometry, starting with the discrete symmetries of polygons
and evolving into the continuous Lie Groups of 3D space: \(SO(3)\) and \(SE(3)\).