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April 25th, 2017, 01:01 PM   #1
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Homework Problem


I am a bit stuck on this question and can't start it.. Any help is very welcome, thanks in advance.

Q: In the ring R=ℚ[x]/(x^5) find the inverse element for (1-2(x^2)+x^3).
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April 25th, 2017, 03:21 PM   #2
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Q[x]/x^5, if I understand your notation, correctly, can be interpreted as the field of all polynomials, with rational coefficients, of degree 5 of less with multiplication defined as regular polynomial multiplication except that we drop all powers of x larger than 4.

To find the multiplicative inverse of 1+ 2x^2+ x^3, we look for ax^4+ bx^3+ cx^2+ dx+ e such that this multiplication gives 1. That is, we look at (x^3+ 2x^2+ 0x+1)(ax^4+ bx^3+ cx^2+ dx+ e)= ax^7+ (2a+ b)x^6+ (2b+ a)x^5+ (a+ 2c+ d)x^4+(c+2e)x^2+ dx+ e= (a+ 2c+ d)x^4+ (b+ 2d+ e)x^3+(c+2e)x^2+ dx+ e since we drop the x^7, x^6, and x^5 terms. We want that to be equal to 1 so we must have a+ 2c+ d= 0, b+ 2d+ e= 0, c+ 2e= 0, d= 0, and e= 1. Solve those five equations for a, b, c, d, and e.
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