Unit 7 – Complex Numbers
... Use this sheet to record a summary of each lesson and as a log of homework assignments that you’ve completed. Homework will be collected at random throughout the unit! Date(s) ...
... Use this sheet to record a summary of each lesson and as a log of homework assignments that you’ve completed. Homework will be collected at random throughout the unit! Date(s) ...
Proofs and Solutions
... four odd numbers, or the product of any ten million odd numbers). There’s a theorem lurking here; find it, state it clearly, and prove it by induction. Theorem. The product of any n ≥ 2 odd numbers is odd. Proof. We already proved the base case in Theorem 2.6 (i.e., we’ve already proven that the pro ...
... four odd numbers, or the product of any ten million odd numbers). There’s a theorem lurking here; find it, state it clearly, and prove it by induction. Theorem. The product of any n ≥ 2 odd numbers is odd. Proof. We already proved the base case in Theorem 2.6 (i.e., we’ve already proven that the pro ...
Rings of Fractions
... Theorem 49. Let R be a commutative ring. Let D be any nonempty subset of R that does not contain 0, does not contain any zero divisors, and is closed under multiplication. Then there exists a commutative ring Q with 1 such that Q contains R as a subring and every element of D is a unit in Q. Theorem ...
... Theorem 49. Let R be a commutative ring. Let D be any nonempty subset of R that does not contain 0, does not contain any zero divisors, and is closed under multiplication. Then there exists a commutative ring Q with 1 such that Q contains R as a subring and every element of D is a unit in Q. Theorem ...
![Rings of constants of the form k[f]](http://s1.studyres.com/store/data/021444599_1-2b48e542456bdb5a68a0329bdee50e0a-300x300.png)
The classification of algebraically closed alternative division rings of
... Baer (see the Introduction of [7]): up to isomorphism, they are the rings of quaternions over real closed fields. In this paper, we extend this result to the nonassociative alternative case. We prove that, up to isomorphism, the algebraically closed nonassociative alternative division rings of finit ...
... Baer (see the Introduction of [7]): up to isomorphism, they are the rings of quaternions over real closed fields. In this paper, we extend this result to the nonassociative alternative case. We prove that, up to isomorphism, the algebraically closed nonassociative alternative division rings of finit ...
What We Need to Know about Rings and Modules
... Definition 2.5 Let R be a commutative ring. Let a, b ∈ R. 1. Then a is a divisor of b, (or a divides b, or a is a factor of b) iff there is c ∈ R so that b = ca. This is written as a | b. 2. b is a multiple of a iff a divides b. That is iff there is c ∈ R so that b = ac. 3. The element b 6= 0 is a p ...
... Definition 2.5 Let R be a commutative ring. Let a, b ∈ R. 1. Then a is a divisor of b, (or a divides b, or a is a factor of b) iff there is c ∈ R so that b = ca. This is written as a | b. 2. b is a multiple of a iff a divides b. That is iff there is c ∈ R so that b = ac. 3. The element b 6= 0 is a p ...
