minimal polynomial of an algebraic element

代数元的极小多项式

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

Translation evidence: Literature usage

Meaning in context

The unique monic polynomial of smallest positive degree in $K[x]$ annihilating an algebraic element $\alpha$.

使代数元取值为零的、次数最小的唯一首一基域多项式。

Example

The minimal polynomial of $\sqrt2$ over $\mathbb Q$ is $x^2-2$.

Usage and distinctions

The minimal polynomial is irreducible; an arbitrary annihilating polynomial need not be.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Wen-Wei Li — Methods in Algebra I, author sourcedefinition def:algebraic-element, nonzero-kernel clause defining the unique monic 极小多项式 of least degree; specialize to an extension field · Checked 2026-10-15

Translation sources

1. 张神星 — 代数数论讲义§1.2.2 整基, statement of Theorem 1.18 uses 极小多项式 · Checked 2026-10-14

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