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Chapter 17: Problem 16

Suppose that \(R\) and \(S\) are isomorphic rings. Prove that \(R[x] \cong S[x]\).

Short Answer

Expert verified
Question: Prove that if two rings \(R\) and \(S\) are isomorphic, their corresponding polynomial rings, \(R[x]\) and \(S[x]\), are also isomorphic. Answer: If two rings R and S are isomorphic, there exists an isomorphism function \(\phi: R \to S\) that preserves addition and multiplication. By defining a function \(f: R[x] \to S[x]\) such that \(f(p(x)) = \phi(a_0) + \phi(a_1)x + \dots + \phi(a_n)x^n\), we can show that f is a bijection, preserves addition, and preserves multiplication. This demonstrates that \(f\) is an isomorphism function between \(R[x]\) and \(S[x]\), proving that if \(R\) and \(S\) are isomorphic, then \(R[x] \cong S[x]\).

Step by step solution

01

Define an isomorphism function between R and S

Let \(\phi: R \to S\) be an isomorphism function between rings \(R\) and \(S\). That is, \(\phi(r_{1} + r_{2}) = \phi(r_{1})+\phi(r_{2})\) and \(\phi(r_{1}r_{2}) = \phi(r_{1})\phi(r_{2})\) for all \(r_{1}, r_{2} \in R\).
02

Define an isomorphism function between R[x] and S[x]

Define a function \(f: R[x] \to S[x]\), such that for a given polynomial \(p(x) = a_0 + a_1x + \dots + a_nx^n \in R[x]\), \(f(p(x)) = \phi(a_0) + \phi(a_1)x + \dots + \phi(a_n)x^n \in S[x]\).
03

Prove that f is a bijection

To show that \(f\) is an isomorphism, we need to prove that \(f\) is a bijection (one-to-one and onto). Since \(\phi\) is a bijection, there is a unique element in \(S\) for each element in \(R\) and vice versa. So, for every polynomial in \(R[x]\), there is a unique polynomial in \(S[x]\). Thus, \(f\) is one-to-one and onto.
04

Prove that f preserves addition

Let \(p(x) = a_0 + a_1x + \dots + a_nx^n\) and \(q(x) = b_0 + b_1x + \dots + b_nx^n\) be two polynomials in \(R[x]\). We need to show that \(f(p(x) + q(x)) = f(p(x)) + f(q(x))\). Now, \((p + q)(x) = (a_0 + b_0) + (a_1 + b_1)x + \dots + (a_n + b_n)x^n\). So, \(f((p+q)(x)) = \phi(a_0 + b_0) + \phi(a_1 + b_1)x + \dots + \phi(a_n + b_n)x^n\). Since \(\phi\) preserves addition, \(f((p+q)(x)) = (\phi(a_0)+\phi(b_0)) + (\phi(a_1)+\phi(b_1))x + \dots + (\phi(a_n) + \phi(b_n))x^n = f(p(x)) + f(q(x))\).
05

Prove that f preserves multiplication

Let \(p(x) = a_0 + a_1x + \dots + a_nx^n\) and \(q(x) = b_0 + b_1x + \dots + b_nx^n\) be two polynomials in \(R[x]\). We need to show that \(f(p(x)q(x)) = f(p(x))f(q(x))\). Now, \((pq)(x) = \sum_{i+j=k} a_ib_j x^k\). So, \(f((pq)(x)) = \sum_{i+j=k}\phi(a_ib_j) x^k\). Since \(\phi\) preserves multiplication, we have \(f((pq)(x)) = \sum_{i+j=k}\phi(a_i)\phi(b_j) x^k\). Notice that this is the product of \(f(p(x))\) and \(f(q(x))\). Therefore, \(f(p(x)q(x)) = f(p(x))f(q(x))\).
06

Conclusion

Since \(f\) is a bijection, and it preserves both addition and multiplication, \(f\) is an isomorphism function between \(R[x]\) and \(S[x]\). Hence, \(R[x] \cong S[x]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Abstract Algebra
Abstract algebra is a significant field within mathematics that focuses on algebraic structures such as groups, rings, and fields.
While a group is defined by a single operation satisfying certain conditions (like closure, associativity, identity, and invertibility), a ring introduces two operations: addition and multiplication.

In understanding rings, it's crucial to identify properties such as commutativity, existence of a multiplicative identity, and the distributivity of multiplication over addition.
Studying different types of rings and how they relate to each other is a fundamental aspect of abstract algebra, providing insights into both theoretical perspectives and practical applications.
  • Abstract algebra allows mathematicians to study the structural properties of mathematical systems abstractly, not tied to specific examples.
  • It lays the groundwork for many other areas in mathematics, including number theory and algebraic geometry.
  • Understanding isomorphisms, which are structure-preserving mappings between algebraic structures, is critical for recognizing when two systems are essentially 'the same' under the hood.
Polynomial Ring
A polynomial ring, often denoted as \(R[x]\), is an extension of a ring \(R\) adding a new element \(x\) that can represent an indeterminate or a variable.
This ring consists of all polynomials with coefficients from \(R\), allowing for the usual operations of addition and multiplication.

Here's an example of operations in a polynomial ring:
  • Addition: \((2x^3 + x) + (3x^2 + 4) = 2x^3 + 3x^2 + x + 4\)
  • Multiplication: \((x + 2) \times (x^2 + x + 1) = x^3 + 3x^2 + 3x + 2\)
These operations have to satisfy the ring axioms, just like the ring from which the coefficients are taken.

Polynomial rings are used in diverse areas such as solving algebraic equations, constructing polynomial functions, and in cryptography, particularly in RSA encryption algorithms.
Ring Homomorphism
A ring homomorphism is a map between two rings that preserves the ring operations of addition and multiplication.
For a map \( \theta: R \to S \), where \(R\) and \(S\) are rings, \( \theta \) is a ring homomorphism if for all elements \( a, b \) in \( R \):
  • \( \theta(a + b) = \theta(a) + \theta(b) \)
  • \( \theta(a \times b) = \theta(a) \times \theta(b) \)
It's important to note that a ring homomorphism doesn't necessarily have to map the multiplicative identity in \( R \) to the multiplicative identity in \(S\), unless we are talking about a ring with unity (where such an identity exists).

Exploring ring homomorphisms helps us understand how structures can be functionally similar and sets the stage for defining isomorphisms, where a bijective (one-to-one and onto) ring homomorphism demonstrates the structural equivalence of rings.

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Most popular questions from this chapter

Let \(p(x)\) and \(q(x)\) be polynomials in \(R[x]\), where \(R\) is a commutative ring with identity. Prove that \(\operatorname{deg}(p(x)+q(x)) \leq \max (\operatorname{deg} p(x), \operatorname{deg} q(x))\).

Cyclotomic Polynomials. The polynomial $$ \Phi_{n}(x)=\frac{x^{n}-1}{x-1}=x^{n-1}+x^{n-2}+\cdots+x+1 $$ is called the cyclotomic polynomial. Show that \(\Phi_{p}(x)\) is irreducible over \(\mathbb{Q}\) for any prime \(p .\)

List all of the polynomials of degree 3 or less in \(\mathbb{Z}_{2}[x]\).

Let \(\mathbb{Q}^{*}\) be the multiplicative group of positive rational numbers. Prove that \(\mathbb{Q}^{*}\) is isomorphic to \((\mathbb{Z}[x],+)\)

Give two different factorizations of \(x^{2}+x+8\) in \(\mathbb{Z}_{10}[x]\).
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