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Using Borevich's system of generators and relations, the classicalArtin-Hasse formula is obtained from scratch in the case when taking $p$-throot of the second argument of the Hilbert symbol gives an unramified$p$-extension of the same degree of irregularity. Under the same assumptions,in the case of Lubin-Tate formal groups, an expression for the Hilbert symbolis obtained in terms of the expansion of elements by the same system ofgenerators.

Artin-Hasse formula for $p^m-$primary elements