{'fraction_non_alphanumeric': 0.03485254691689008, 'fraction_numerical': 0.002680965147453083, 'mean_word_length': 4.151724137931034, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Stickelberger Condition on Cyclic Galois Extensions

Let R be a commutative ring, C a finite abelian group, S a Galois extension of R with group C, in the sense of [1]. Viewing S as an RC-module defines the Picard invariant map [4] from the Harrison group Gal (R, C) of isomorphism classes of Galois extensions of R with group C to CI (RC), the class group of RC. The image of the Picard invariant map is known to be contained in the subgroup h Cl (RC) of primitive elements of CI (RC) (for definition see below). Characterizing the image of the Picard invariant map has been of some interest, for the image describes the extent of failure of Galois extensions to have normal bases. Let R be the ring of integers of an algebraic number field K.