1.
True or False: The numbers 83 and 62 belong to the same coset of \(\Z/\langle 7 \rangle\text{.}\)
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Worksheet Polynomial Quotient Rings
Let \(R\) be a commutative ring with unity. Recall that \(R[x]\) is the ring of all polynomials in the variable \(x\) with coefficients in the ring \(R\text{.}\) The goal of this activity is to explore what the quotient rings look like when we start with a polynomial ring.
Throughout this activity, we will consider the quotient ring \(A = \mathbb Q[x]/\langle x-2\rangle\text{.}\) Note that \(\langle x-2\rangle\) is the ideal generated by the polynomial \(x-2\text{.}\) It will be helpful to compare this to the more familiar \(\Z/\langle 7 \rangle \cong \Z_7\text{.}\)
2.
True or False: the polynomials \(2x^2 + 3x + 5\) and \(x^2 + 4x +7\) belong to the same coset of \(A\) (i.e., they are congruent modulo \(x-2\)).
3.
Evaluate both polynomials above at \(x = 2\) (that is, substitute 2 for \(x\)). What do you notice?
4.
To what coset of \(\Z/\langle 7 \rangle\) does the integer 2487 belong?
5.
To what coset of \(A\) does the polynomial \(x^3 + 5x^2 - 3x + 1\) belong? Of course, you could answer \(x^3 + 5x^2 - 3x + 1 + \langle x-2\rangle\text{,}\) but can you do better? What does “better” even mean here?
6.
7.
What familiar ring is \(A = \Q[x]/\langle x-2\rangle\) isomorphic to? Hint: consider a homomorphism from \(\Q[x]\) which makes \(\langle x-2\rangle\) the kernel.
