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Solvable Groups
Solvable Groups

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MATH UN3025 - Midterm 2 Solutions 1. Suppose that n = p · q is the

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... Proof. Suppose that f (x) ∈ R[x] is primitive in R[x] and irreducible in F [x]. If f (x) = a(x)b(x) in R[x], then one of a(x) and b(x) must be a unit in F [x], so of degree 0. Suppose without loss of generality that a(x) = a0 ∈ R. Then a0 divides all coefficients of f (x), and, because f (x) is prim ...
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... It can be shown that any principal ideal domain is a Dedeking ring, but the converse is not true. However, we will see in the next section that most rings we are concerned with are Dedekind (cf. Proposition (3.2)). We require one last definition. Definition 2.6. Let R be an integral domain and K its ...
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Root of unity



In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.
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