Math 361, Spring 2017, Assignment 10
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Relative algebraic closure (of a field $F$ in an extension $E$).
- $\overline{\mathbb{Q}}$ (a.k.a. the field of algebraic numbers).
- Algebraically closed field.
- Prime subfield (of a field).
- Characteristic (of a field).
- $GF(p^n)$.
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem characterizing algebraically closed fields (i.e. giving several conditions equivalent to the property of being algebraically closed).
- Fundamental Theorem of Algebra.
- Theorem concerning degrees of irreducible polynomials in $\mathbb{C}[x]$.
- Theorem concerning degrees of irreducible polynomials in $\mathbb{R}[x]$.
- Theorem restricting the cardinality of a finite field.
- Theorem concerning existence of finite fields of certain cardinalities.
- Theorem relating finite fields of equal cardinality.
Carefully describe the following procedures:[edit]
- Procedure to factor polynomials in $\mathbb{C}[x]$.
- Procedure to factor polynomials in $\mathbb{R}[x]$.
- Procedure to construct finite fields.
Solve the following problems:[edit]
- Section 33, problems 1, 2, and 3.
- 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.)
- 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.)
- Factor the polynomial $x^3 - 1$ over $\mathbb{R}$.
- 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 $\overline{\mathbb{Q}}$.
Questions[edit]
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)