Field extension

Simple extension

A field extension 𝐿 :𝐾 is called simple iff 𝐿 is generated by the adjunction of a single element, field i.e. 𝐿 =𝐾(𝜗) for some 𝜗 ∈𝐿.¹ Such a 𝜗 is called a primitive element.

Classification

Let 𝐾(𝜗) :𝐾 be a simple extension, and consider the evaluation map

𝜖:𝐾[𝑡]→𝐾[𝜗]⊆𝐾(𝜗)𝑓(𝑡)↦𝑓(𝜗)

Then

1. If 𝜖 is injective then 𝐾(𝜗) :𝐾 is an infinite extension, whence 𝐾(𝜗) is isomorphic to the field of rational functions 𝐾(𝑡);
2. If 𝜖 is not injective then 𝐾(𝜗) :𝐾 is an extension of finite degree 𝑛, and

𝐾(𝜗)≅𝖥𝗅𝖽𝐾[𝑡]⟨𝑚𝜗(𝑡)⟩

where 𝑚𝜗(𝑡) ∈𝐾[𝑡] is the minimal polynomial of 𝜗.

Proof

By the First isomorphism theorem, the image of 𝜖 is isomorphic to 𝐾[𝑡]/ker⁡𝜖. Since 𝐾(𝜗) is an integral domain, so is 𝐾[𝜗], and thus by Condition for a quotient commutative ring to be an integral domain, ker⁡𝜖 must be a prime ideal in 𝐾[𝑡].

First, consider ker⁡𝜖 =0, i.e. 𝜖 is injective. By the universal property for the Field of fractions, 𝜖 extends to a unique homomorphism

¯𝜖:𝐾(𝑡)→𝐾(𝜗)

where 𝐾(𝑡) ≅¯𝜖(𝐾(𝑡)) ≤𝐾(𝜗) is a field containing 𝐾 and 𝜗, whence by definition ¯𝜖(𝐾(𝑡)) =𝐾(𝜗). By injectivity, {𝜗𝑖}∞𝑖=0 are linearly independent so we have an infinite extension.

Now consider 𝔞 =ker⁡𝜖 ≠0. Since 𝐾[𝑡] is a Euclidean domain and thus a PID, it follows 𝔞 =⟨𝑝(𝑡)⟩ for a unique monic irreducible nonconstant polynomial 𝑝(𝑡) ∈𝐾[𝑡]. Since ⟨𝑝(𝑡)⟩ is maximal in 𝐾[𝑡], the image of 𝜖 is a subfield containing 𝜗 =𝜖(𝑡), and by the same token as above we have 𝜖(𝐾(𝑡)) =𝐾(𝜗), giving the claimed isomorphism.

Properties

1. If 𝐾(𝛼) and 𝐾(𝛽) have the same minimal polynomial 𝑓(𝑥) ∈𝐾[𝑥], then there exists a unique isomorphism of field extensions 𝜑 :𝐾(𝛼) →𝐾(𝛽) such that 𝜑(𝛼) =𝛽.
2. More generally, an isomorphism of ground fields 𝜓 :𝐾1 →𝐾2 such that 𝜓(𝑚𝛼(𝑥)) =𝑚𝛽(𝑥) lifts to an isomorphism of extensions ¯𝜓 :𝐾1(𝛼) →𝐾2(𝛽).

Proof of 1–2

Note that an isomorphism of simple field extensions is completely determined by the image of the primitive element.

Now using the isomorphism described in Classification,

𝐾(𝛼)≅𝐾[𝑥]⟨𝑓(𝑥)⟩≅𝐾(𝛽)

which necessarily fixes 𝐾 and maps 𝛼 ↦𝛽, proving ^P1, whereof ^P2 is a straightforward generalization.

Results

• Bound on the automorphism group of a finite simple extension.
• Simplicity of an algebraic extension
• By the Primitive element theorem, every finite separable extension is simple.

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Footnotes

1. 2009. Algebra: Chapter 0, §VII.1.2, pp. 387–388 ↩