GRE Math Problem #57: Subring of R[x] From Real Numbers

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Discussion Overview

The discussion revolves around a GRE math problem concerning the identification of subrings within the ring of polynomials R[x] with coefficients in the real numbers R. Participants explore the definitions and properties that qualify a subset of R[x] as a subring, focusing on specific examples provided in the problem.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant proposes that all three subsets listed in the problem could be considered subrings, based on their properties.
  • Another participant questions whether the subset of polynomials of even degree has a multiplicative identity, suggesting that this is a necessary condition for being a subring.
  • A third participant clarifies that only an additive identity is required for a subset to be a subring, which the subset of even degree polynomials possesses.
  • Further discussion reveals that definitions of "ring" may vary, with some sources distinguishing between rings with and without unity, which could affect the interpretation of the problem.
  • Participants acknowledge the importance of understanding these definitions for the GRE and note the potential for confusion stemming from different conventions.

Areas of Agreement / Disagreement

Participants express differing views on the requirements for a subset to be classified as a subring, particularly regarding the necessity of a multiplicative identity. The discussion remains unresolved as to whether the second subset qualifies as a subring under all definitions.

Contextual Notes

The discussion highlights the ambiguity in definitions of rings and subrings, particularly concerning the presence of a multiplicative identity, which may vary among mathematical texts.

jammidactyl
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I'm reviewing the practice booklet for the GRE and came across a question I can't solve. Problem #57 for reference.

http://www.ets.org/Media/Tests/GRE/pdf/Math.pdf

Let R be the field of real numbers and R[x] the ring of polynomials in x with coefficients in R. Which of the following subsets of R[x] is a subring of R[x]?

I. All polynomials whose coefficient of x is zero.
II. All polynomials whose degree is an even integer, together with the zero polynomial.
III. All polynomials whose coefficients are rational numbers.

I figured the answer was "all of the above", but the answer in the back says just I and III.

If you add or subtract two polynomials of even degree, you get another polynomial of even degree or the zero polynomial. If you multiply two polynomials of even degree, the answer also is a polynomial of even degree. Since it's a subset and satisfies these conditions, isn't II a subring?

I think I'm making a really simple mistake with some obvious counterexample.
 
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Does II have a multiplicative identity?
 
It only needs an additive identity to be a subring, which it has.
 
Apparently it depends on which definition of the term "ring" you're used to! This is good information to know for the test... never realized there was such a difference.
 

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