Totally Ramified Primes and Eisenstein Polynomials
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چکیده
is Eisenstein at a prime p when each coefficient ci is divisible by p and the constant term c0 is not divisible by p 2. Such polynomials are irreducible in Q[T ], and this Eisenstein criterion for irreducibility is the way nearly everyone first meets Eisenstein polynomials. Here, we will show Eisenstein polynomials are closely related to total ramification of primes in number fields. Let K be a number field, with degree n over Q. A prime number p is said to be totally ramified in K when pOK = p n. For example, in Z[i] we have (2) = (1 + i)2, so 2 is totally ramified in Q(i). The link between Eisenstein polynomials and totally ramified primes is described in the following two theorems, which are converses of each other.
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تاریخ انتشار 2013