ABSTRACT

Given a ring R and a polynomial f with coefficients in R, how many solutions does the equation f(x) = 0 have in R? The answer depends on both f and R; that is, even the same polynomial f can have a different number of solutions depending on a change in the ring R. Let’s look at a hopefully familiar example followed by a less familiar one.

Consider the quadratic polynomial equation x 2 − 3 = 0, and note that coefficients are in the integers Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_1.jpg"/> , but also in the real numbers R https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_2.jpg"/> and in the complex numbers C https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_3.jpg"/> . We can see that in both R https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_4.jpg"/> and C https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_5.jpg"/> , there are two solutions ± 3 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_6.jpg"/> , but in Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_7.jpg"/> there are no solutions. On the other hand, the equation x 2 + 3 = 0 has no solutions in both Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_8.jpg"/> and R https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_9.jpg"/> , but it has two solutions ± i 3 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_10.jpg"/> in C https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_11.jpg"/> . Note, however, that by the Quadratic Formula (Example 1.1) in these three rings any quadratic polynomial equation can have at most two solutions. We shall be able to generalize this idea nicely in Chapter 35.

Consider the quadratic polynomial equation x 2 + 6x + 8 = 0 and note that the coefficients can be viewed as coming from Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_12.jpg"/> or from Z 12 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_13.jpg"/> , the integers modulo 12. By factoring, the equation has two solutions in Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_14.jpg"/> : −2 and −4. However, in Z 12 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429324819/ca52079e-ce0e-4a28-aa2a-95d630084a0c/content/inline-math28_15.jpg"/> , there actually are four solutions: 2, 4, 8 and 10 (check!!). Hence it is possible in certain rings for a quadratic polynomial equation to have more than two solutions.