## Galois cohomology and cyclic and abelian extensions

Last update: 01 February 2012

## Galois cohomology and cyclic and abelian extensions

Let $𝔽$ be a field.

• A cyclic extension of $𝔽$ is a Galois extension $𝔼$ of $𝔽$ such that $\mathrm{Gal}\left(𝔼/𝔽\right)$ is cyclic.
• An abelian extension of $𝔽$ is a Galois extension $𝔼$ of $𝔽$ such that $\mathrm{Gal}\left(𝔼/𝔽\right)$ is abelian.
• An abelian extension of exponent dividing $n$ is an abelian extension $𝔼$ of $𝔽$ such that ${g}^{n}=1$ for all $g\in \mathrm{Gal}\left(𝔼/𝔽\right)$.

(Hilbert's Theorem 90) Let $𝔼$ be a cyclic extension of $𝔽$ and let $\sigma$ be a generator of $\mathrm{Gal}\left(𝔼/𝔽\right)$.

1. (a) Let $x\in {𝔼}^{*}$.
2. (a') Let $x\in {𝔼}^{*}$. If there exists an element $y\in {𝔼}^{*}$ such that $x=\frac{y}{\sigma \left(y\right)}$ then it is unique up to multiplication by elements of ${𝔽}^{*}$.
3. (b) Let $x\in 𝔼$.
4. (b') Let $x\in 𝔼$. If there exists an element $z\in 𝔼$ such that $x=z-\sigma \left(z\right)$ then it is unique up to adding elements of $𝔽$.

(Kummer theory) Assume $𝔽$ contains a primitive ${n}^{th}$ root of unity.

1. There is a bijection
2. If $H\supseteq {𝔽}^{*}$ is a subgroup such that $H\supseteq {\left({𝔽}^{*}\right)}^{n}$ then and is a basis of $𝔽\left({H}^{\frac{1}{n}}\right)$ over $𝔽$.

Let $𝔽$ be a field with $\mathrm{char}\left(𝔽\right)=0$ and let incomplete sentence

## Notes and References

Where are these from?

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