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Tensor product of fields

In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield.

Version
v1 · 2026-09-28 · History
Domain-specific #
12488
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Field Theory, Commutative Algebra → Mathematics

Core Idea

Tensor product of fields is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield.

Scope of Application

  • Examples. with K the field of rational functions in the indeterminate T over the finite field with p elements (see Separable polynomial: the point here is that P is not separable).

  • Compositum of fields. Either one starts in a situation where an ambient field is easy to identify (for example if K and L are both subfields of the complex numbers), or one proves a.

  • Compositum of fields. The idea behind the compositum is to make the smallest field containing two other fields.

  • Compositum of fields. In order to formally define the compositum, one must first specify a tower of fields.

  • Compositum of fields. Let k be a field and L and K be two extensions of k.

Clarity

A clear use of Tensor product of fields names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield.

Manages Complexity

Tensor product of fields compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—(This type of result can be verified, in general, by using the ramification theory of algebraic number theory.).—and the practical consequence—one can prove this by calculating the dimension of the tensor product over \mathbb{Q} as 9, and observing that the splitting field does contain two (indeed three).

Abstract Reasoning

  1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield.
  3. Check operation and conditions. The structure of the ring can be analysed by considering all ways of embedding both K and L in some field extension of N.
  4. Demand recognition evidence. via the map induced by 1\mapsto (1,1), z\mapsto (\sqrt{2},-\sqrt{2}) . 5.

Knowledge Transfer

Within the home domain. Knowledge about Tensor product of fields transfers literally when a new case preserves the same carrier type, relation, and recognition test. with K the field of rational functions in the indeterminate T over the finite field with p elements (see Separable polynomial: the point here is that P is not separable). Either one starts in a situation where an ambient field is easy to identify (for example if K and L are both subfields of the complex numbers), or one proves a result that allows one to place both K and L (as isomorphic copies) in.

Relationships to Other Abstractions

Local relationship map for Tensor product of fieldsParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Tensor productof fieldsDOMAINDomain-specific abstraction: Algebra over a Ring — is a kind ofAlgebraover a RingDOMAIN

Current abstraction Tensor product of fields Domain-specific

Parents (1) — more general patterns this builds on

  • Tensor product of fields is a kind of Algebra over a Ring Domain-specific

    The tensor product of fields over a common subfield is an algebra over that subfield, though it need not remain a field.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Tensor product of fields sits in a crowded region of the domain-specific corpus (38th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Structures, Groups & Operators (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08