splitting field

分裂域

Algebra & Algebraic Geometry · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For f∈K[x]f\in K[x], an extension generated over KK by all roots of ff in which ff splits into linear factors.

由给定多项式全部根在基域上生成、使其完全分解为一次因子的扩域。

Example

Q(2)\mathbb Q(\sqrt2) splits x2−2x^2-2 over Q\mathbb Q.

Usage and distinctions

An extension in which a polynomial splits need not be its splitting field if it contains unnecessary elements.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Wen-Wei Li — Methods in Algebra I, author source分裂域和正规扩张 (sec:splitting-field), opening definition naming 分裂域, nonconstant-polynomial family and root-generation clauses; singleton-family specialization · Checked 2026-10-15

Translation sources

  1. 李文威 — 代数学方法 卷一，作者源文件分裂域和正规扩张 (sec:splitting-field), opening definition naming 分裂域, nonconstant-polynomial family and root-generation clauses; singleton-family specialization · Checked 2026-10-15
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