By Napalkov V. V.

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**Extra info for A basis in the space of solutions of a convolution equation**

**Example text**

Is any simple ER-algebra with center (i. , an embedding (A,A)-bimodule. 3 F = Z(A), A Let be a simple 9 C A be a maximal let the ring of integers. 7L- lor Im(nrA) and F* If R C F be 0 F* are described as follows: Im(nr,,,) and nrA(K1(A)) = FTM (ii) and let pp-order, is a %-algebra, then A If (t) Qp-algebra with center or Then nrA : K1(A) is injectiue; and 43 STRUCTURE THEOREMS FOR K, OF ORDERS CHAPTER 2. A nrl(K1(IR)) = R*. is a ID-algebra, then set = {u E F* : v(u) > 0 for all ramified v: F '- Itl; R + = F+ fl R*.

For some unique (K = @) Gal(Q'n/tD) K n > 1 such that each (Z/n)*: a E (Z/n)*. I'n. Let by a primitive n-th root of unity. (ZJn)*, Gal(KCn/K) T E Cal(Kcn/K) can be has the form Furthermore: and Zfn C QCn In particular, under the identification maximal Z-order in ([C] (ii) K is an abelian Galois extension, and KC /K identified as a subgroup of then and K, denote a field extension of D[Cn] is the ring of ndInnd' the ndInZCd- is a finite group, and is a splitting field for G: i. , KCn[G] char(K)4'exp(G)In, is a product of 26 BASIC ALGEBRAIC BACKGROUND CHAPTER 1.

We r > A 0 the section by noting following the more specialized properties of p-adic group rings. 10 Q, and let R C F Fix a prime p, let F be the ring of integers. be any finite extension of BASIC ALGEBRAIC BACKGROUND CHAPTER 1. 4(v). integers in If p C R for each R. /F for some is a R/p[Cn] so pRi is unramified. is the The last i. of characteristic zero, a cyclotomic algebra is a twisted group ring of the form finite cyclotomic extension of A i, Fi = FCn statement follows since over p = char(R/p)Fn), by are the rings of R.