James Lynch MAT 501 Class Notes: 10/29/09 Some Exam Problems
... and b in a ring if: d|a and d|b and whenever there exists c in the ring such that c|a and c|b we have that c|d (ii) d=-2 is a greatest common divisor of 4 and 6 in the integers because d clearly divides both 4 and 6, and if we have some c that also divides 4 and 6 then c can only be equal to one of ...
... and b in a ring if: d|a and d|b and whenever there exists c in the ring such that c|a and c|b we have that c|d (ii) d=-2 is a greatest common divisor of 4 and 6 in the integers because d clearly divides both 4 and 6, and if we have some c that also divides 4 and 6 then c can only be equal to one of ...
MTE-06-2008
... Let I, J be ideals of a ring R such that I + J = R. Prove that I J = IJ and if IJ = 0, then R ; R R . ...
... Let I, J be ideals of a ring R such that I + J = R. Prove that I J = IJ and if IJ = 0, then R ; R R . ...
Computer Security - Rivier University
... a n a a a (i.e. applied n-1 times) a -n (a' ) n , where a' is the inverse of a • A group G is cyclic if every element of G is a power gk (k is an integer) of a fixed element g G. The element g is said to generate the group, or to be a generator of the group. • A cyclic group is alway ...
... a n a a a (i.e. applied n-1 times) a -n (a' ) n , where a' is the inverse of a • A group G is cyclic if every element of G is a power gk (k is an integer) of a fixed element g G. The element g is said to generate the group, or to be a generator of the group. • A cyclic group is alway ...
Finite Fields
... Example 1.1. Constructing F8 . The polynomial P (X) = X 3 + X + 1 is irreducible over F2 , otherwise it would have a factor of degree 1, i.e., a root in F2 , while P (0) = P (1) = 1. Then, F23 can be represented by all triples (a, b, c) of elements in F2 , or equivalently by all polynomials of the f ...
... Example 1.1. Constructing F8 . The polynomial P (X) = X 3 + X + 1 is irreducible over F2 , otherwise it would have a factor of degree 1, i.e., a root in F2 , while P (0) = P (1) = 1. Then, F23 can be represented by all triples (a, b, c) of elements in F2 , or equivalently by all polynomials of the f ...
Field _ extensions
... -/consequence of the methods used, in the 1920's and 1930's, . to generalize the theory to arbitrary fields. From this viewpoint the central object of study ceases to be a polynomial, and becomes instead - a 'field extension' related to a polynomial. Every polynomial f over a field K defines -anothe ...
... -/consequence of the methods used, in the 1920's and 1930's, . to generalize the theory to arbitrary fields. From this viewpoint the central object of study ceases to be a polynomial, and becomes instead - a 'field extension' related to a polynomial. Every polynomial f over a field K defines -anothe ...
pdf file
... IV. (IPA) Integer Parts that are models of PA FACT: [Exponentiation on the non-negative elements of a model of PA] The graph of the exponential function 2y = z on N is definable by an L-formula, and P A proves the basic properties of exponentiation. Thus any model of P A is endowed with an exponent ...
... IV. (IPA) Integer Parts that are models of PA FACT: [Exponentiation on the non-negative elements of a model of PA] The graph of the exponential function 2y = z on N is definable by an L-formula, and P A proves the basic properties of exponentiation. Thus any model of P A is endowed with an exponent ...
1. Rings and Fields
... 1.1. Introduction to Rings. The operations of addition and multiplication in real numbers have direct parallels with operations which may be applied to pairs of integers, pairs of integers mod another positive integer, vectors in Rn , matrices mapping Rn to Rm , polynomials with real or integer coef ...
... 1.1. Introduction to Rings. The operations of addition and multiplication in real numbers have direct parallels with operations which may be applied to pairs of integers, pairs of integers mod another positive integer, vectors in Rn , matrices mapping Rn to Rm , polynomials with real or integer coef ...
Here - UCSD Mathematics - University of California San Diego
... 0 6= b ∈ R such that ba = 0 (resp. ab = 0). a is called a zero divisor if there are non-zero elements b and b0 such that ab = b0 a = 0. Example 9. If a ∈ U (R), then a is not a left (or right) zero divisor. Definition 10. Let R be a commutative unital ring. It is called an integral domain if it has ...
... 0 6= b ∈ R such that ba = 0 (resp. ab = 0). a is called a zero divisor if there are non-zero elements b and b0 such that ab = b0 a = 0. Example 9. If a ∈ U (R), then a is not a left (or right) zero divisor. Definition 10. Let R be a commutative unital ring. It is called an integral domain if it has ...
The Reals
... Properties of Binary Operations Identity elements: If there exists an element i ∈ S such that for all a ∈ S, i ⊙ a = a ⊙ i = a, then we say that i is an identity element for ⊙. Example: For S = ℝ, 0 is the identity element for addition and 1 is the identity element for multiplication. Inverse eleme ...
... Properties of Binary Operations Identity elements: If there exists an element i ∈ S such that for all a ∈ S, i ⊙ a = a ⊙ i = a, then we say that i is an identity element for ⊙. Example: For S = ℝ, 0 is the identity element for addition and 1 is the identity element for multiplication. Inverse eleme ...
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