6.6. Unique Factorization Domains
... 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 ...
... 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 ...
Self-study Textbook_Algebra_ch9
... have to find out the length of its side first. It is same as finding a number whose square is equal to 9. We have two answers, namely 32 = 9 , and (−3) 2 = 9 . Since length of a side cannot be a negative number, we determine that the length of the side of this square table is 3 m. In general, if the ...
... have to find out the length of its side first. It is same as finding a number whose square is equal to 9. We have two answers, namely 32 = 9 , and (−3) 2 = 9 . Since length of a side cannot be a negative number, we determine that the length of the side of this square table is 3 m. In general, if the ...
(pdf)
... 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 ...
... 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 ...
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.