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Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$. Then $ U $ is a subgroup of $ L^{\times} $ = the set of all non zero elements in $L$. Now what is the index of the subgroup $ U $ of $ L^{\times} $?

Note: By Hilbert's theorem 90 we have $ U =\{\sigma(a)/a : a\in L^{\times}\} $. So one can consider the map $ \phi : L^{\times} \rightarrow L^{\times} $ by $ \phi(a) = \sigma(a)/a $. Then $ L^{\times}/ \mathbb{Q} \cong U $.

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$. Then $ U $ is a subgroup of $ L^{\times} $ = the set of all non zero elements in $L$. Now what is the index of the subgroup $ U $ of $ L^{\times} $?

Note: By Hilbert's theorem 90 we have $ U =\{\sigma(a)/a : a\in L^{\times}\} $. So one can consider the map $ \phi : L^{\times} \rightarrow L^{\times} $ by $ \phi(a) = \sigma(a)/a $. Then $ L^{\times}/ \mathbb{Q}^\times \cong U $.

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$. Then $ U $ is a subgroup of $ L^{\times} $ = the set of all non zero elements in $L$. Now what is the index of the subgroup $ U $ of $ L^{\times} $?

Note: By Hilbert's theorem 90 we have $ U =\{\sigma(a)/a : a\in L^{\times}\} $. So one can consider the map $ \phi : L^{\times} \rightarrow L^{\times} $ by $ \phi(a) = \sigma(a)/a $. Then $ L^{\times}/ \mathbb{Q} \cong U $. So we have to study $ L^{\times}/\mathbb{Q} $ .

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$. Then $ U $ is a subgroup of $ L^{\times} $ = the set of all non zero elements in $L$. Now what is the index of the subgroup $ U $ of $ L^{\times} $?

Note: By Hilbert's theorem 90 we have $ U =\{\sigma(a)/a : a\in L^{\times}\} $. So one can consider the map $ \phi : L^{\times} \rightarrow L^{\times} $ by $ \phi(a) = \sigma(a)/a $. Then $ L^{\times}/ \mathbb{Q} \cong U $.

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$. Then $ U $ is a subgroup of $ L^{\times} $ = the set of all non zero elements in $L$. Now what is the index of the subgroup $ U $ of $ L^{\times} $?

Note: By Hilbert's theorem 90 we have $ U =\{\sigma(a)/a : a\in L^{\times}\} $. So one can consider the map $ \phi : L^{\times} \rightarrow L^{\times} $ by $ \phi(a) = \sigma(a)/a $. Then $ L^{\times}/ U \cong \operatorname{Im}(\phi) $. So we have to study $ \operatorname{Im}(\phi) $.

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$. Then $ U $ is a subgroup of $ L^{\times} $ = the set of all non zero elements in $L$. Now what is the index of the subgroup $ U $ of $ L^{\times} $?

Note: By Hilbert's theorem 90 we have $ U =\{\sigma(a)/a : a\in L^{\times}\} $. So one can consider the map $ \phi : L^{\times} \rightarrow L^{\times} $ by $ \phi(a) = \sigma(a)/a $. Then $ L^{\times}/ \mathbb{Q} \cong U $. So we have to study $ L^{\times}/\mathbb{Q} $ .