Math 361, Spring 2014, Assignment 3

Carefully define the following terms, then give one example and one non-example of each:

1. Monic polynomial.
2. Monic generator (of a principal ideal).
3. Field extension.
4. Extension field.
5. Base field.
6. Injection (associated with a field extension).

Carefully state the following theorems (you need not prove them):

1. Classification of prime ideals in $F[x]$.
2. Kronecker's Theorem.

Solve the following problems:

1. Consider the polynomial $f = x^3 - 2$ in $\mathbb{Q}[x]$.
   • Show that $f$ is irreducible over $\mathbb[Q]$.
   • Now consider the extension field $E_1 = \mathbb{Q}[x]/\langle f\rangle$. Is $f$ still irreducible when regarded as a polynomial with coefficients in $E_1$? If not, then write an explicit non-trivial factorization of $f$ over $E_1$.
   • Does $f$ split over $E_1$? (Warning: this is a hard question. In principle you can answer it now, with a lot of work, but soon we will learn an efficient way to answer it. If you prefer, take a guess at this and forge ahead.) If not, can you produce another extension $E_2$ over which it does split?

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Monic Polynomial

   A monic polynomial is a polynomial whose leading term is 1.

   Example:

   \(x^4 + 5x^2 + 7\)

   Non-example:

   \(5x^2\)

2. Monic Generator of a Principal Ideal

   Given an ideal \(\langle p \rangle\) of \(F[x]\), we can find a monic polynomial \(q\) such that \(\langle p \rangle = \langle q \rangle\).

   Example:

   The monic generator of the ideal \(\langle 5x^3 + 6x\rangle\) is \(x^3 + 6/5x\).

   Non-example:

   \(10x^3 + 12x\) is not a monic generator of the same ideal. Is it a generator, but it isn't monic.

3. Field Extension

   A field extension is a triple \((F,E,\iota)\) such that \(F\) and \(E\) are fields, and \(\iota:F\rightarrow E\) is a monomorphism.

   Example:

   \(\mathbb{Q},\mathbb{R},\iota\) is a field extension, where \(\iota(x)=x\).

   Non-example:

   \((\mathbb{Z},\mathbb{Q},\iota)\) is not a field extension, because \(\mathbb{Z}\) is not a field.

4. Extension Field

   Given a field extension \((F,E,\iota)\), \(E\) is the extension field.

   Example:

   In the above example, \(\mathbb{R}\) is the extension field.

   Non-example:

   \(\mathbb{Q}\) isn't.

5. Base Field

   Given a field extension \((F,E,\iota)\), \(F\) is the base field.

   Example:

   In the above example, \(\mathbb{Q}\) is the base field.

   Non-example:

   \(\mathbb{R}\) isn't.

6. Injection of a Field Extension

   \(\iota\).

Theorems:

1. Classification of Prime Ideals in \(F[x]\)

   An ideal of \(F[x]\) is prime if and only if it is generated by an irreducible polynomial. Remember that all ideals of \(F[x]\) are generated by a single element.

2. Kronecker's Theorem

   Given a polynomial \(p\) in \(F[x]\), we can find a field extension \(F,E,\iota\) such that \(p\) has a root in \(E\).

Solutions:

1. \(f\) is irreducible over \(\mathbb{Q}\) because it is a cubic with no roots.

   So \(\alpha\) is a root of \(x^3-2\) over \(\mathbb{Q}/\langle f \rangle\). After all, \(\alpha^3 -2=0\) mod \(\langle f \rangle\). To actually factor the polynomial, we need to divide \(x^3-2\) by \(\alpha\) within the polynomial ring \(\mathbb{Q}/\langle f \rangle[x]\). The result is \(x^2+\alpha x + \alpha^2\). It's too hard to typeset the actual division, so I won't try. Just remember that \(\alpha^3=2\) in \(\mathbb{Q}/\langle f \rangle\).

   This polynomial does not split over \(E_1\). \(x^3-2\) doesn't even split over \(\mathbb{R}\), and the field we've created is a "subset" of \(\mathbb{R}\).