Worksheet Extending Fields to Factor
Goal: Build the smallest field possible in which \(p(x) = x^3 + 3x^2 - x + 2\) is NOT irreducible.
Note that \(p(x)\) is irreducible over \(\Q\) because it has no roots in \(\Q\) (why is this and why is that enough?). So let’s invent a new number, call it \(\s\text{,}\) and insist that \(\s\) is a root of \(p(x)\text{.}\) Then consider the smallest field \(E\) larger than \(\Q\) that also contains \(\s\text{.}\)
2.
The element \(\s^3\) is in \(E\text{,}\) but this can also be written using smaller powers of \(\s\text{.}\) How?
3.
Describe \(E\) as a set using set builder notation. In other words, \(E\) is the set of all elements of the form …
4.
Wait: why are we doing this? Our goal is for \(p(x)\) to factor. Does it? What would one of the factors be?
5.
Wait again: we want \(E\) to be a field. Is it? What would we need to check?
6.
List five elements in the quotient ring \(\Q[x]/\langle p(x)\rangle\) (using the same \(p(x)\) from the previous page). Remember, these will all be cosets.
7.
The element \(x^3+\langle p(x) \rangle\) is an element of \(\Q[x]/\langle p(x) \rangle\text{,}\) but it can also be written as a “simpler” coset. How?
8.
Describe \(\Q[x]/\langle p(x) \rangle\) as a set using set builder notation. In other words, this quotient ring is the set of all cosets of the form …
9.
Wait: if we want to show that \(E\) is a field, and \(E\) is basically the same as \(\Q[x]/\langle p(x) \rangle\text{,}\) then we could just show \(\Q[x]/\langle p(x) \rangle\) is a field. What would this mean? What do we need to verify?
