PlanetMath: Hilbert Theorem 90

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Hilbert Theorem 90

(Theorem)

Let be a finite Galois extension with Galois group $G = \operatorname{Gal}(L/K)$ . The modern formulation of Hilbert's Theorem 90 states that the first Galois cohomology group is 0.

The original statement of Hilbert's Theorem 90 differs somewhat from the modern formulation given above, and is nowadays regarded as a corollary of the above fact. In its original form, Hilbert's Theorem 90 says that if is cyclic with generator $\sigma$ , then an element $x \in L$ has norm 1 if and only if

$\displaystyle x = y/\sigma(y)$

for some $y \in L$ . Note that elements of the form $y/\sigma(y)$ are obviously contained within the kernel of the norm map; it is the converse that forms the content of the theorem.

"Hilbert Theorem 90" is owned by djao.

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Other names:

Hilbert's Theorem 90, Satz 90

Attachments:: stronger Hilbert theorem 90 (Theorem) by alozano
proof of Hilbert Theorem 90 (Proof) by mathcam

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Cross-references: converse, map, kernel, contained, norm, generator, cyclic, group, Galois cohomology, Galois group, Galois extension, finite
There are 5 references to this entry.

This is version 3 of Hilbert Theorem 90, born on 2002-01-07, modified 2008-02-07.
Object id is 1435, canonical name is HilbertTheorem90.
Accessed 5971 times total.

Classification:

AMS MSC:	11R34 (Number theory :: Algebraic number theory: global fields :: Galois cohomology)
	11S25 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Galois cohomology)
	11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)

Pending Errata and Addenda

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