Multiplicative group of field extension over base field

Asked May 28 '20 at 14:40

Active May 28 '20 at 17:48

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Let $k\subset K$ a field extension. Assume that $K$ is infinite. Then the index of $k^*$ in $K^*$, $[K^*:k^*]$ (as groups) cannot be finite. How can I prove this?

If $k$ is finite then I know how to solve this since, if $\mid k\mid =q$ then , if we assume that $[K^*:k^*] = n$ is not finite, we have that $K$ can be partitioned into $n$ disjoint sets with $q$ elements. So $\mid K\mid = nq$ not finite, therefore contradiction.

I dont know how to prove it for $k$ is infinite as well.

edited May 28 '20 at 15:39

asked May 28 '20 at 14:40

Popescu Daniel

3

I see no question in your post. – amWhy May 28 '20 at 14:41
In the case that $[K:k]<\infty$, why don’t you consider the Norm from $K$ down to $k$? – Lubin May 28 '20 at 18:03
1

Or let $a \in K \setminus k$ and consider the elements $a + b$ with $b \in k$. – Derek Holt May 28 '20 at 18:45
4

One way of seeing this is that if the index were finite, then $K$ would be the union of finitely many (1-dimensional) subspaces. But that is impossible. – Jyrki Lahtonen May 28 '20 at 19:10

Multiplicative group of field extension over base field

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