## Exercises 17.8 Sage Exercises

### 1.

Consider the polynomial \(x^3-3x+4\text{.}\) Compute the most thorough factorization of this polynomial over each of the following fields: (a) the finite field \({\mathbb Z}_5\text{,}\) (b) a finite field with 125 elements, (c) the rationals, (d) the real numbers and (e) the complex numbers. To do this, build the appropriate polynomial ring, and construct the polynomial as a member of this ring, and use the `.factor()`

method.

### 2.

â€śConway polynomialsâ€ť are irreducible polynomials over \({\mathbb Z}_p\) that Sage (and other software) uses to build maximal ideals in polynomial rings, and thus quotient rings that are fields. Roughly speaking, they are â€ścanonicalâ€ť choices for each degree and each prime. The command `conway_polynomial(p, n)`

will return a database entry that is an irreducible polynomial of degree \(n\) over \({\mathbb Z}_p\text{.}\)

Execute the command `conway_polynomial(5, 4)`

to obtain an allegedly irreducible polynomial of degree 4 over \({\mathbb Z}_5\text{:}\) \(p = x^{4} + 4x^{2} + 4x + 2\text{.}\) Construct the right polynomial ring (i.e., in the indeterminate \(x\)) and verify that `p`

is really an element of your polynomial ring.

First determine that p has no linear factors. The only possibility left is that `p`

factors as two quadratic polynomials over \({\mathbb Z}_5\text{.}\) Use a list comprehension with *three* `for`

statements to create *every* possible quadratic polynomial over \({\mathbb Z}_5\text{.}\) Now use this list to create every possible product of two quadratic polynomials and check to see if `p`

is in this list.

More on Conway polynomials is available at Frank LĂĽbeck's site^{â€‰16â€‰}.

### 3.

Construct a finite field of order \(729\) as a quotient of a polynomial ring by a principal ideal generated with a Conway polynomial.

### 4.

Define the polynomials \(p = x^3 + 2x^2 + 2x + 4\) and \(q = x^4 + 2x^2\) as polynomials with coefficients from the integers. Compute `gcd(p, q)`

and verify that the result divides both `p`

and `q`

(just form a fraction in Sage and see that it simplifies cleanly, or use the `.quo_rem()`

method).

PropositionÂ 17.10 says there are polynomials \(r(x)\) and \(s(x)\) such that the greatest common divisor equals \(r(x)p(x)+s(x)q(x)\text{,}\) *if the coefficients come from a field*. Since here we have two polynomials over the integers, investigate the results returned by Sage for the extended gcd, `xgcd(p, q)`

. In particular, show that the first result of the returned triple is a multiple of the gcd. Then verify the â€ślinear combinationâ€ť property of the result.

### 5.

For a polynomial ring over a field, every ideal is principal. Begin with the ring of polynomials over the rationals. Experiment with constructing ideals using two generators and then see that Sage converts the ideal to a principal ideal with a single generator. (You can get this generator with the ideal method `.gen()`

.) Can you explain how this single generator is computed?

`www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol`