Question

The notion of transcendence degree is analogous to the idea of the dimension of a vector space. If \(k \subset K\), we say that \(x_{1}, \ldots, x_{n} \in K\) are algebraically independent if there is no nonzero polynomial \(F \in k\left[X_{1}, \ldots, X_{n}\right]\) such that \(F\left(x_{1}, \ldots, x_{n}\right)=0 .\) By methods entirely analogous to those for bases of vector spaces, one can prove: (a) Let \(x_{1}, \ldots, x_{n} \in K, K\) a finitely generated extension of \(k\). Then \(x_{1}, \ldots, x_{n}\) is a minimal set such that \(K\) is algebraic over \(k\left(x_{1}, \ldots, x_{n}\right)\) if and only if \(x_{1}, \ldots, x_{n}\) is a maximal set of algebraically independent elements of \(K .\) Such \(\left\\{x_{1}, \ldots, x_{n}\right\\}\) is called a transcendence basis of \(K\) over \(k\). (b) Any algebraically independent set may be completed to a transcendence basis. Any set \(\left\\{x_{1}, \ldots, x_{n}\right\\}\) such that \(K\) is algebraic over \(k\left(x_{1}, \ldots, x_{n}\right)\) contains a transcendence basis. (c) tr. \(\operatorname{deg}_{k} K\) is the number of elements in any transcendence basis of \(K\) over \(k\).

Short answer

#Answer# The transcendence degree, denoted as tr. deg\(_k K\), is the number of elements in any transcendence basis of the field extension \(K\) over \(k\), where a transcendence basis is a set of algebraically independent elements over which the entire field extension is algebraic. Any algebraically independent set may be completed to a transcendence basis, and any set such that \(K\) is algebraic over it contains a transcendence basis.

Step-by-step solution

(a) Show equivalence of terms

We have the following definitions: 1. \(x_1, \ldots, x_n\) is a minimal set such that \(K\) is algebraic over \(k(x_1, \ldots, x_n)\). 2. \(x_1, \ldots, x_n\) is a maximal set of algebraically independent elements of \(K\). We need to show that these two definitions are equivalent. (\(\Rightarrow\)) Suppose that \(x_1, \ldots, x_n\) is a minimal set such that \(K\) is algebraic over \(k(x_1, \ldots, x_n)\). If it were not a maximal set of algebraically independent elements, we could find another element \(x_{n+1} \in K\) such that the set \(x_1, \ldots, x_{n+1}\) is algebraically independent. Then, \(K\) would be algebraic over \(k(x_1, \ldots, x_{n+1})\), contradicting the minimality of the original set. (\(\Leftarrow\)) Conversely, suppose that \(x_1, \ldots, x_n\) is a maximal set of algebraically independent elements of \(K\). Then, \(K\) is algebraic over \(k(x_1, \ldots, x_n)\), otherwise there would exist another algebraically independent element, contradicting the maximality of the original set. Therefore, the definitions are equivalent, and such a set is called a transcendence basis of \(K\) over \(k\).

(b) Completing and finding a transcendence basis

We have to prove two properties: 1. Any algebraically independent set may be completed to a transcendence basis. 2. Any set \(\{x_1, \ldots, x_n\}\) such that \(K\) is algebraic over \(k(x_1, \ldots, x_n)\), contains a transcendence basis. Property 1: Take any algebraically independent set \(S\) in \(K\). If \(K\) is algebraic over \(k(S)\), then \(S\) is already a transcendence basis, and we are done. Otherwise, there exists some element \(x \in K\) which is not algebraic over \(k(S)\). We can add this element to our set and check whether \(K\) is algebraic over this new set. We can continue this process until \(K\) is algebraic over our set, obtaining a transcendence basis. Property 2: Let \(\{x_1, \ldots, x_n\}\) be a set such that \(K\) is algebraic over \(k(x_1, \ldots, x_n)\). We can remove elements of this set if they are algebraic over the smaller set, and we repeat this process until we get an algebraically independent set. This remaining set is a transcendence basis, as it is minimal and \(K\) would still be algebraic over this smaller set.

(c) Definition of transcendence degree

The transcendence degree, denoted as tr. deg\(_k K\), is the number of elements in any transcendence basis of the field extension \(K\) over \(k\). Since we showed that any set \(\{x_1, \ldots, x_n\}\) such that \(K\) is algebraic over \(k(x_1, \ldots, x_n)\) contains a transcendence basis, we can use any transcendence basis to determine the transcendence degree.

Key concepts

Algebraically Independent

When discussing the concept of algebraically independent elements within a field extension, it's similar to thinking about linearly independent vectors in a vector space. An element, or a set of elements, is considered algebraically independent over a field, if there exists no nonzero polynomial with coefficients in that field for which these elements can be substituted to yield zero.

Imagine having a set of numbers \(x_1, x_2, ..., x_n\) from a larger field \(K\) containing a smaller field \(k\). These numbers are algebraically independent over \(k\) if we can't find a polynomial—where coefficients come from \(k\) and variables are replaced by \(x_1, x_2, ..., x_n\)—that would equate to zero, unless the polynomial is the zero polynomial itself. It's a way of saying these elements bring new 'dimensions' to \(K\) that could not be expressed by any combination of elements purely from \(k\).

Understanding algebraic independence is key to grasping more advanced concepts like transcendence bases and transcendence degree, which play vital roles in both field theory and algebraic geometry.

Transcendence Degree

The transcendence degree is an essential characteristic when exploring field extensions. It tells us the 'size' of the extension in terms of algebraically independent elements—a concept reflective of the dimension of a vector space. Specifically, the transcendence degree of a field extension \(K/k\) is defined as the maximum number of algebraically independent elements over \(k\) that you can find in \(K\).

Using the step-by-step solution given, we understand that the transcendence degree, written as \(\text{tr. deg}_k K\), equals the number of elements in any transcendence basis of \(K\) over \(k\). This number remains consistent, regardless of which transcendence basis we choose, just like the dimension of a vector space is invariant of the choice of the basis. Therefore, calculating the transcendence degree is a clever way to understand the 'complexity' or the 'capacity' that the extension \(K\) has over \(k\), in terms of new elements not constructible from \(k\) alone.

Field Extension

A field extension is a very intuitive concept. Imagine you have a box (which is our field \(k\)) filled with a set of elements (numbers) obeying certain algebraic rules. An extension of this field, denoted as \(K/k\), is like a larger box (field \(K\)) containing all the original elements and possibly more. But these new elements have to also respect the same type of algebraic rules—the field axioms.

Field extensions allow mathematicians to introduce new elements that might not be solvable within the smaller field, such as square roots or other radicals, and transcendental numbers like \(\pi\) or \(e\). This idea is pivotal in solving equations and understanding the structure of different mathematical systems. And as shown in the textbook exercise, recognizing algebraically independent sets and transcendence bases can be crucial in fully understanding the structure of such field extensions.

Algebraic Geometry

Algebraic geometry might sound daunting, but it's a fascinating study that bridges algebra with geometry. The fundamental objects examined in this field are the solutions to polynomial equations, which are then considered as geometric entities. For example, the classic parabola is the geometric representation of a quadratic equation, which we encounter frequently even outside of mathematical contexts.

The connection to our problem lies in the fact that algebraic geometry often deals with field extensions, as solutions to polynomial equations may not always reside within the original field. By leveraging concepts like algebraically independent elements and transcendence bases, algebraic geometers can navigate between abstract algebraic structures and concrete geometric visualizations. Transcendence bases, for instance, help define the 'dimension' of an algebraic variety within the field of algebraic geometry, analogous to the way they aid in defining the transcendence degree in field theory.