Tensor product of fields

Tensor Product of Fields

The tensor product of fields is a concept in abstract algebra, particularly within the realms of ring theory and field theory. It extends the idea of a tensor product from vector spaces to fields, which are algebraic structures with more stringent properties. Understanding the tensor product of fields requires a grasp of both the tensor product in linear algebra and the nature of fields in algebra.

Definition

Given two fields, \(F\) and \(K\), over a common subfield \(E\), the tensor product of \(F\) and \(K\) over \(E\), denoted \(F \otimes_E K\), is the vector space generated by the symbols \(f \otimes k\), where \(f \in F\) and \(k \in K\), subject to the relations that make the operation compatible with field addition and multiplication, as well as scalar multiplication by elements of \(E\). Formally, the tensor product \(F \otimes_E K\) is a quotient of the free abelian group generated by the set \(F \times K\) by the subgroup generated by all elements of the form:

1. \((f+f', k) - (f, k) - (f', k)\), 2. \((f, k+k') - (f, k) - (f, k')\), 3. \((af, k) - a(f, k)\), 4. \((f, ak) - a(f, k)\), 5. \((ff', k) - (f, k)(f', k)\), 6. \((f, kk') - (f, k)(f, k')\),

for all \(f, f' \in F\), \(k, k' \in K\), and \(a \in E\).

Properties

The tensor product \(F \otimes_E K\) itself may not always form a field. One of the key issues is that the tensor product may contain zero divisors, making it impossible to define multiplicative inverses for all nonzero elements, a requirement for field structures. However, under certain conditions, such as when \(F\) and \(K\) are both finite extensions of \(E\) and \(F \cap K = E\), the tensor product can be embedded into a larger field, leading to interesting applications in Galois theory and the study of field extensions.

Applications

The concept of tensor products of fields finds applications in various areas of mathematics, including:

- **Algebraic Geometry**: In the study of algebraic varieties, tensor products of fields can be used to understand the function fields of varieties over different bases. - **Number Theory**: The tensor product appears in the context of number fields and their Galois groups, providing insights into the arithmetic properties of fields. - **Quantum Mechanics**: In a more applied context, the mathematical framework of quantum mechanics uses tensor products to describe the state spaces of composite systems.

Challenges and Further Reading

The construction and study of tensor products of fields pose several challenges, primarily due to the potential for zero divisors and the lack of a field structure in the tensor product itself. For readers interested in exploring this topic further, it is recommended to delve into textbooks and articles on advanced abstract algebra, focusing on chapters concerning ring theory, field theory, and linear algebra.