Math 361, Spring 2014, Assignment 5

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

1. Vector space (over a field $F$).
2. Spanning set.
3. Linearly independent set.
4. Basis.
5. Dimension.
6. Algebraic extension.
7. Algebraically closed field.
8. Algebraic closure.

Carefully state the following theorems:

1. Theorem concerning the existence of bases.
2. Dimension formula.
3. Theorem relating finite-dimensional extensions to algebraic extensions.
4. Theorem concerning the existence of algebraic closures.

Solve the following problems:

1. Section 30, problems 1, 4, 5, 7, and 9.
2. Section 31, problems 1, 3, 5, 23, and 24.

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

Solutions:

Definitions:

1. Vector space (over a field $F$).

   Let $F$ be a field. A vector space over F is a triple $(V,+,\mu)$ where $(V,+)$ is an abelian group (with identity $\vec{0}$) and $\mu:F\times V\rightarrow V$ satisfies:

   $$(c_1+c_2)\vec{v} = c_1\vec{v} +c_2\vec{v}$$ $$c(\vec{v_1}+\vec{v_2}) = c\vec{v_1} + c\vec{v_2}$$ $$ c_1(c_2\vec{v}) = (c_1c_2)\vec{v}$$ $$1\cdot\vec{v} = \vec{v}$$

   Note: we abbreviate $\mu (c,\vec{v})$ by $c\vec{v}$

2. Spanning set.

   Let $S$ be any subset of the vector space $V$. $S$ spans $V$ if every element of $V$ is a linear combination.

3. Linearly independent set.

   $S$ is a linearly independent if no element of $S$ is a linear combination of other elements.

4. Basis.

   $S$ is a basis for $V$ is if it spans and is linearly independent.

5. Dimension.

   The dimension of $V$ over $F$ is the cardinality of any basis written $dim_F V$ or $\left[V:F\right]$

6. Algebraic extension.

   The extension $F\rightarrow E$ is algebraic is every element in $E$ is algebraic over $F$.

   Example:

   $\mathbb{C}$ is algebraic $/\mathbb{R}$

   Non-example

   $\mathbb{R}$ is not algebraic $/\mathbb{Q}$

7. Algebraically closed field.

   A field $F$ is algebraically closed if it satisfies any of the following:

   1. Every non-constant polynomial in $F\left[ x\right]$ has a root in $F$.
   2. Every non-constant polynomial in $F\left[ x\right]$ splits over $F$.
   3. Every irreducible polynomial in $F\left[ x\right]$ has degree 1.
   4. Every algebraic extension of $F$ is trivial.
   Note: These are equivalent statements, so if $F$ satisfies one of them, it satisfies all of them,

   Example:

   $\mathbb{C}$ <\p>

   Non-example:

   $\mathbb{R}$ is not algebraically closed since $x^2+1$ is not constant but has no roots in $\mathbb{R}$.

8. Algebraic closure.

   An extension $F\rightarrow\Omega$ is said to be an algebraic closure of $F$ if:

   1. it is algebraic
   2. $\Omega$ is algebraically close

   Example:

   $\mathbb{R}\rightarrow\mathbb{C}$ is an algebraic closure of $\mathbb{R}$.

   Non-example

   $\mathbb{Q}\rightarrow\mathbb{C}$ is not an algebraic closure of $\mathbb{Q}$, since it is not an algebraic extension.

Theorems:

1. Theorem concerning the existence of bases.

   Every vector space has a basis

   or

   Any linearly independent set can be grown to a basis and any spanning set can be shrunk to a basis.

2. Dimension formula.

   Suppose we have a chain of extensions

   $$F\rightarrow K\rightarrow E$$

   Then $\left[E:F\right] = \left[E:K\right]\left[K:F\right]$.

3. Theorem relating finite-dimensional extensions to algebraic extensions.

   The extension $F\rightarrow E$ is a finite extension $\Leftrightarrow$ it is algebraic and finitely generated.

4. Theorem concerning the existence of algebraic closures.
   1. Every field has an algebraic extension
   2. Universal mapping property of an algebraic closure: If $F\rightarrow \bar{F}$ is an algebraic closure, and $F\rightarrow E$ is any algebraic extension, then there exists a monomorphism $E\rightarrow\bar{F}$.
   3. If $\Omega$ is any algebraically closed extension of $F$, then there exists a unique $\phi:\bar{F}\rightarrow\Omega$