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Let $L/K$ be a Galois extension of number fields with Galois group $G$. Wediscuss a new method to obtain elements in $\mathbb{Z}[G]$ which annihilate theclass group of $L$. Using this method, we obtain annihilators of class groupsof cyclotomic fields. We show that these annihilators are new. Some moreconsequences are also discussed. Moreover, we mention some results andconnections to highlight importance of primes of higher residue degree.

Primes of higher degree and Annihilators of Class groups