Show that 𝐑[x] /⟨x^2+1⟩is a field.

Name: Problem 30, Chapter 14: Contemporary Abstract Algebra
Uploaded: 2019-10-13T03:00:17-08:00
Duration: 5 min 28 s
Channel: Bryan Lynn
Description: Show that 𝐑[x] /⟨x^2+1⟩is a field.

Key Concepts

Field

A field is a set equipped with two operations, addition and multiplication, in which every nonzero element has a multiplicative inverse. Demonstrating that R[x]/?x²+1? is a field shows that the quotient ring has a structure where division (excluding division by zero) is possible, which is a foundational concept in algebra and field theory.

Quotient Ring

A quotient ring is constructed by partitioning a ring into equivalence classes determined by an ideal. In the case of polynomial rings, forming the quotient R[x]/?x²+1? involves treating two polynomials as equivalent if their difference is a multiple of x²+1. This construction is essential in creating new algebraic structures and is used to prove that certain quotient rings are fields when the ideal is maximal.

Irreducible Polynomial

An irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials over the same field. In this context, x²+1 is irreducible over R because it has no real roots, and its irreducibility is critical in ensuring that the ideal generated by it is maximal.

Polynomial Ring

A polynomial ring, such as R[x], is the set of all polynomials with coefficients in R and with the usual operations of addition and multiplication. It provides the structure needed to perform algebraic operations with polynomials, serving as the foundation for constructing more complex algebraic objects like quotient rings and field extensions.

Maximal Ideal

A maximal ideal is an ideal that is as large as possible without being equal to the entire ring, meaning there are no other ideals strictly between it and the entire ring. The property that an ideal is maximal ensures that the quotient of the ring by this ideal is a field, providing a powerful tool for constructing fields from rings.

Transcript

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