Math 361, Spring 2017, Assignment 10

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

1. Relative algebraic closure (of a field $F$ in an extension $E$).
2. $\overline{\mathbb{Q}}$ (a.k.a. the field of algebraic numbers).
3. Algebraically closed field.
4. Prime subfield (of a field).
5. Characteristic (of a field).
6. $GF(p^n)$.

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

1. Theorem characterizing algebraically closed fields (i.e. giving several conditions equivalent to the property of being algebraically closed).
2. Fundamental Theorem of Algebra.
3. Theorem concerning degrees of irreducible polynomials in $\mathbb{C}[x]$.
4. Theorem concerning degrees of irreducible polynomials in $\mathbb{R}[x]$.
5. Theorem restricting the cardinality of a finite field.
6. Theorem concerning existence of finite fields of certain cardinalities.
7. Theorem relating finite fields of equal cardinality.

Carefully describe the following procedures:

1. Procedure to factor polynomials in $\mathbb{C}[x]$.
2. Procedure to factor polynomials in $\mathbb{R}[x]$.
3. Procedure to construct finite fields.

Solve the following problems:

1. Section 33, problems 1, 2, and 3.
2. Construct a field with $125$ elements. (That is, describe how to write down elements of your field, give rules for addition and multiplication together with examples of these calculations, and explain why all this results in a field. You do not need to write down complete addition or multiplication tables.)
3. Factor the polynomial $x^3 - 1$ over $\mathbb{C}$. (Hint: the roots all lie on the unit circle in the complex plane, and they are equally spaced around it. See the Wikipedia article on "Roots of unity" for more information.)
4. Factor the polynomial $x^3 - 1$ over $\mathbb{R}$.
5. Show that $\overline{\mathbb{Q}}$ is algebraically closed, as follows: let $p=a_0+\dots+a_nx^n$ be any non-constant polynomial with coefficients in $\overline{\mathbb{Q}}$.

(a) Explain why $p$ must have a root in $\mathbb{C}$. Choose such a root, and call it $\beta$.

(b) Using the Dimension Theorem, show that $\mathbb{Q}(a_0,\dots,a_n)$ (i.e. the smallest subfield containing $\mathbb{Q}$ and the coefficients of $p$) must be finite-dimensional over $\mathbb{Q}$.

(c) Using the classification of simple extensions, show that $\mathbb{Q}(a_0,\dots,a_n,\beta)$ is finite-dimensional over $\mathbb{Q}(a_0,\dots,a_n)$.

(d) Using the Dimension Theorem a second time, conclude that $\beta$ is algebraic over $\mathbb{Q}$, and hence lies in $\overline{\mathbb{Q}}$.

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

Questions

1)Theorem characterizing algebraically closed fields (i.e. giving several conditions equivalent to the property of being algebraically closed):1)Every non-constant is in F[x] has a root. 2)Every non-constant in f[x] factors as a product of linear factors ("splits") 4) Every irreducible in F[x] has a degree 1. 5)Every algbrica extension of F is trivial.

2)Fundamental Theorem of Algebra: $\mathbb{C}$ is algebraically closed $x^5 +x^4+\pi(x^3)+ (e+i)x^2 +2x)$ has roots in $\mathbb{C}$

3)Theorem concerning degrees of irreducible polynomials in $\mathbb{C}[x]$: ??? Could not find at all so sad.

Any irreducible element of $\mathbb{C}[x]$ has degree one. -Steven.Jackson (talk) 11:51, 15 May 2017 (EDT)

4)Theorem concerning degrees of irreducible polynomials in $\mathbb{R}[x]$: Any irr element $\mathbb{R}[x]$ has degree 1 or 2.

5)Theorem restricting the cardinality of a finite field: Suppose $F$ is a finite field then, $|F|$is a power of a prime.

6)Theorem concerning existence of finite fields of certain cardinalities:?? sort of idea. For every prime power $p^n$ there is a field $F$ with $|F|$=p.

Should be "...there is a field $F$ with $|F|=p^n$. -Steven.Jackson (talk) 11:51, 15 May 2017 (EDT)

7)Theorem relating finite fields of equal cardinality: Any two finite filed with the same cardinality are in fact isomorphic.

My question is are my theorems correct? But more important I could not find theorem 3 and not sure completely on theorem 6. Can you make sure the others are correct as well.

They are correct with the exceptions noted above. -Steven.Jackson (talk) 11:51, 15 May 2017 (EDT)

I would like to know what the procedure to construct finite fields are AS WELLL AS #3?

To construct a field with $p^n$ elements, use the Sieve to find an irreducible polynomial $f\in\mathbb{Z}_p[x]$ of degree $n$. (One is guaranteed to exist. If $n$ is large, then the Sieve is impractical, but there are more efficient methods that we haven't learned about in this class.) Then put $F=\mathbb{Z}_p[x]/\langle f\rangle$. This will be a field with $p^n$ elements. (Although there will usually be more than one possible $f$, the resulting field is independent of this choice, up to isomorphism.)

To factor $x^3-1$ over $\mathbb{C}$, first note that $1$ is a root, so $x-1$ must be a factor. Using long division, one finds that $x^3-1 = (x-1)(x^2+x+1)$. To find the roots of the quadratic factor, use the quadratic formula; it turns out that $\frac{-1\pm i\sqrt{3}}{2}$ are roots. So finally one obtains the factorization $x^3-1 = (x-1)\left(x-\frac{-1+i\sqrt{3}}{2}\right)\left(x-\frac{-1-i\sqrt{3}}{2}\right)$. -Steven.Jackson (talk) 11:51, 15 May 2017 (EDT)