Abstract Algebra II by Randall R. Holmes

By Randall R. Holmes

Show description

Read or Download Abstract Algebra II PDF

Best linear books

The Symmetric Eigenvalue Problem (Classics in Applied Mathematics)

A droll explication of concepts that may be utilized to appreciate a few of an important engineering difficulties: these facing vibrations, buckling, and earthquake resistance. whereas containing significant thought, this can be an utilized arithmetic textual content that reads as though you're eavesdropping at the writer conversing out loud to himself.

Signal Enhancement with Variable Span Linear Filters

This ebook introduces readers to the radical idea of variable span speech enhancement filters, and demonstrates the way it can be utilized for powerful noise aid in a number of methods. extra, the ebook presents the accompanying Matlab code, permitting readers to simply enforce the most principles mentioned. Variable span filters mix the tips of optimum linear filters with these of subspace equipment, as they contain the joint diagonalization of the correlation matrices of the specified sign and the noise.

Extra info for Abstract Algebra II

Example text

For convenience, we will restrict our search to integral domains. The theorem can be thought of as saying that prime numbers are the building blocks for the integers greater than one. We need a generalization of these building blocks to our arbitrary integral domain. The reader most likely learned that a prime number is an integer greater than one having as positive factors only one and itself. This definition yields 2, 3, 5, 7, 11, and so on. The words “greater than” and “positive” here create obstacles for any generalization, since the axioms for an integral domain provide no notions of order or positivity.

Ii) Let a ∈ F . Since a0 + a0 = a(0 + 0) = a0, cancellation gives a0 = 0. (iii) Let a ∈ F and v ∈ V . Since av + (−a)v = (a + (−a))v = 0v = 0 (using part (i)), we have (−a)v = −av. 4 Subspace Let F be a field and let V be a vector space over F . A subspace of V is a subgroup of (V, +) that is closed under scalar multiplication. Thus, a subset W of V is a subspace if and only if 62 (i) 0 ∈ W , (ii) w, w ∈ W ⇒ w + w ∈ W , (iii) w ∈ W ⇒ −w ∈ W , (iv) a ∈ F, w ∈ W ⇒ aw ∈ W. We write W ≤ V to indicate that W is a subspace of V .

1 Theorem. If R is a UFD, then the polynomial ring R[x] is a UFD. Proof. ) Here are some applications of the theorem: • Since Z is a UFD, so is Z[x]. • A field F is a UFD, so F [x] is a UFD as well. 6). • If R is a UFD, then so is the ring of polynomials over R in n indeterminants R[x1 , x2 , . . , xn ] (n ∈ N) defined recursively by putting R[x1 , x2 , . . , xn ] ∼ = R[x1 , x2 , . . , xn−1 ][xn ]. This claim follows immediately from the theorem by using induction on n. 8 Induced homomorphism of polynomial rings Let R and R be commutative rings with identity and let σ : R → R be a homomorphism.

Download PDF sample

Rated 4.61 of 5 – based on 49 votes