5.1 General Field Theory

Know basic definitions: Characteristic, prime field; field extension $E / F$ (simple, algebraic, transcendental, finite, normal, separable, inseparable); degree $[E: F] ; F(S), F[S]$ for subsets $S$ of $E / F$. Minimum polynomial of an algebraic element $a \in E / F$; degree of $a$ over $F$. Splitting field of a polynomial over $F$.

Know: Transitivity of degree $[K: F]=[K: E][E: F]$ if $K / E / F$;
$[K(\alpha): K]=\operatorname{deg}\left(\mu_{\alpha \mid K}\right)$ if $\alpha$ is algebraic over $K$ with minimal polynomial $\mu_{\alpha \mid K}$.
Know how to construct field extensions of F : simple algebraic extension $E=F[x] /(p(x))$ for an irreducible polynomial $p(x)$ in $F[x]$ : this is a simple algebraic extension of degree $[E: F]= \operatorname{deg}(p)$, generated by the image of $x$ with minimal polynomial $p$. Simple transcendental extension: $E=F(x)=$ field of rational functions in $x=$ field of fractions of $F[x]$.

Know about splitting fields: Existence and uniqueness of splitting fields; connection with normal extensions: a finite field extension $L / K$ is normal iff $L$ is the splitting field of some (nonconstant) polynomial $f \in K[x]$; extension lemma for field isomorphisms.

Know about algebraically closed fields: any field $K$ has an algebraic extension $\bar{K}$ which is algebraically closed, called the algebraic closure of $K$; we have in particular $\overline{\mathbb{R}}=\mathbb{C}$ (FUNDAMENTAL THEOREM OF ALGEBRA).

Know about separable polynomials and field extensions: definition of separability; the basic criterion: $f \in K[x]$ is separable iff $\operatorname{gcd}(f, D(f))=1$; consequence: fields of characteristic 0 and finite fields are perfect, i.e. all their algebraic extensions are automatically separable.

Primitive Element Theorem: If $L / K$ is finite and separable, then $L$ is a simple extension of $K$, i.e. there exists $\alpha \in L$ such that $L=K(\alpha)$. (One can also show that $L$ is a simple extension of $K$ iff there are only finitely many intermediate fields between $K$ and $L$.)

KNOW HOW TO: construct all finite fields $\mathbb{F}_{q}\left(q=p^{e}\right.$, split $x^{p^{e}}-x$ over $\left.\mathbb{F}_{p}=\mathbb{Z}_{p}\right)$.
Avoid the following mistakes: If $M / L$ and $L / K$ are normal field extensions, $M / K$ need not be normal; if $\operatorname{ch} \operatorname{ar}(K)=0$, not all polynomials in $K[x]$ are separable (but the irreducible polynomials in $K[x]$ are); if $f \in K[x]$ for some field $K$ satisfies $D(f) \neq 0$, this does not imply that $f$ is separable (it does if $f$ is irreducible); a polynomial in $K[x]$ which doesn’t have any roots in $K$ need not be irreducible (it is if $\operatorname{deg}(f) \leq 3$ ).

Related Problems

$F$ is an arbitrary field.

1. (Apr $77 \# 2$ ) If $F$ is finite SHOW
   (a) $|F|=q$ is a power of a prime $p$,
   (b) $F$ splits $x^{q}-x$ over $\mathbb{Z}_{p}$,
   (c) $F^{*}$ is cyclic of order $q-1$.

2. (Apr $77 \# 4 \mathrm{~b}$ ) If $p(x) \in F[x]$ is irreducible, SHOW there is a finite extension $E / F$ containing a root of $p(x)$.

3. (Nov 77 #3) Give an example of an inseparable extension $E / \mathbb{Q}$ of degree 7 .

If you don’t succeed, give an arbitrary example of an algebraic but inseparable field extension.
4. (Nov 77 #4) Define finite extension and algebraic extension; does either imply the other?
5. (May 78 #10) Give a polynomial whose splitting field is a field of 9 elements; repeat for 18 elements.
6. (Sep 82 #4) If $E / F$ is algebraic and each $a \in E$ belongs to a normal sub-extension $E_{a}(F \subseteq \left.E_{a} \subseteq E\right)$, show $E / F$ is normal.
7. (Sep $83 \# 3$ ) (a) Show any finite subgroup of $F^{*}$ (the multiplicative group of invertible elements of $F$ ) must be cyclic. (Done in class but you might recall the argument.)
(b) Give an example of a finite nonabelian group contained in $R^{*}$ for a ring $R$.
8. (Feb $84 \# 6$ ) If $[F(a): F]$ is odd show $F\left(a^{2}\right)=F(a)$.
9. (Sep $84 \# 7$ ) Factor $x^{8}-1$ into irreducibles in $\mathbb{Z}_{p}[x]$ for all primes $p$. (Clarification: You should describe the prime factorization of $x^{8}-1$ in $\mathbb{Z}_{p}[x]$ for any given prime number $p$ (distinguish cases!) but you needn’t compute explicitly, as elements of $\mathbb{Z}_{p}$, those coefficients of the (monic) irreducible factors which are $\neq 0, \pm 1$.)
10. (1985 #3b) Show the set of all $2 \times 2$ matrices $\left(\begin{array}{rr}a & b \\ \lambda b & a\end{array}\right)$ over $\mathbb{F}_{7}$ form (under the usual matrix addition and multiplication) a ring of size 49 . For which $\lambda \in \mathbb{F}_{7}$ is this a field?
11. (Sep $86 \# 1$ ) If $F=\mathbb{F}_{7}$, show $p(x)=x^{2}+1$ and $p(x)=x^{3}+x+1$ are irreducible in $F[x]$, and show $F[x] /(p(x))$ are fields (give their cardinalities).
12. (Jan $87 \# 4$ ) If $F \subseteq K \subseteq E$ with $K / F$ finite, show that if $K / F$ is separable (resp. normal, Galois), then also $K(a) / F(a)$ is separable (resp. normal, Galois) for any $a \in E$.
13. (May $89 \# 3$ ) If $a, b$ are algebraic over $F$ of degrees $m, n$, show $[F(a, b): F] \leq m n$, with equality if $m, n$ are relatively prime. Give an example where the inequality is strict.
14. (Aug 89 #3) Show the set of all $2 \times 2$ matrices $\left(\begin{array}{rr}a & b \\ 2 b & a\end{array}\right)$ over $\mathbb{F}_{5}$ form (under the usual matrix addition and multiplication) a field of size 25 .
15. (Aug $89 \# 5$ ) If $[E: F]$ is finite, show any ring endomorphism of $E$ which fixes $F$ is an automorphism of $E$.
16. (Aug $94 \# 6$ ) Show that $F=\mathbb{F}_{2}[x] /\left(x^{4}+x+1\right)$ is a field of characteristic 2 containing a primitive 15 -th root of unity. Exhibit such a root.
17. (Jan $95 \# 3$ ) Let $K$ and $L$ be finite extensions of a field $F$, both contained in a field $E$. Let $K L$ be the set of finite sums of products of members of $K$ and $L$. Explain why $K L$ is a subring of $E$ that is finite-dimensional over $F$. From that, show that $K L$ is a field and, in fact, the smallest subfield of $E$ containing $K$ and $L$.
18. (Aug $97 \# 3$ ) Let $K(x)$ be the field of rational functions over the field $K$. Prove Lüroth’s Theorem: for any nonconstant function $f \in K(x)$ the degree $[K(x): K(f)]$ is finite. Can you describe $[K(x): K(f)]$ in terms of $f$ ?
19. (Jan 98 #6) The finite field $\mathbb{F}_{32}$ of 32 elements can be constructed as the extension $\mathbb{F}_{2}(\beta)$ where $\beta$ is a root of the polynomial $x^{5}+x^{2}+1$ in $\mathbb{F}_{2}[x]$. Find the minimal polynomial of $\beta^{3}$ over $\mathbb{F}_{2}$.
20. (Aug $99 \# 5$ ) Let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\left(a_{n} \neq 0\right)$ be a complex polynomial of degree $n>1$, and $f^{\prime}$ its derivative. Let $\alpha_{1}, \ldots, \alpha_{n}$ be the $n$ roots of $f$ and $\alpha_{1}^{\prime}, \ldots, \alpha_{n-1}^{\prime}$ the $n-1$ roots of the derivative (listing each root as many times as its multiplicity). Show that the average of the roots of $f$ equals the average of the roots of $f^{\prime}$.
(Hint: Use the relations between the coeffiecients of a polynomial and the roots of that polynomial.)
21. (Aug 99 #8) If $F$ is finite, show that every element $\alpha \in F$ is the sum $\alpha=\beta_{1}^{2}+\beta_{2}^{2}$ of two squares (for some $\beta_{1}, \beta_{2} \in F$ ).
22. (Aug $99 \# 9$ ) If $p$ is a prime number congruent to $1 \bmod 8$, show that 2 is a "quadratic residue" $\bmod p$, i.e. there is an integer $a$ such that $a^{2} \equiv 2$ modulo $p$. (Hint: show there is an $\varepsilon \in \mathbb{Z}_{p}$ with $\varepsilon^{4}=-1$, and consider $\alpha=\varepsilon+\varepsilon^{-1}$.)
23. (May 03, #8) Let $K=\mathbb{Z} / p \mathbb{Z}$ be the field of $p$ elements (where $p$ is a prime). For an integer $d>0$, we let

$$\sigma_{d}=\sum_{x \in K} x^{d}$$

Show that $\sigma_{d}=-1$ if $p-1$ divides $d$ and $\sigma_{d}=0$ otherwise.
24. (Aug 05 #4) Let $K$ be a $\mathbb{Q}$-subalgebra of $M_{n}(\mathbb{Q})$. If $K$ is a field, prove that $[K: \mathbb{Q}] \leq n$.
25. (Jan 06 # 9) Prove that $\sqrt{2}+\sqrt[3]{3}$ is irrational.
26. (Jan $08 \# 7$ ) Let $F$ be a field.
(a) Show that if $a \in E$ is an element of a field extension of $F$ with $[F(a): F]=7$, then $F\left(a^{3}\right)=F(a)$.
(b) Show that any subgroup of order 8 of the multiplicative group $F^{\times}$of the field $F$ must be cyclic. Is this also true of subgroups of order 16 ?
27. (Aug 08 #3) Prove or disprove the following statements about irreducibility.
(a) If $\operatorname{Gal}(E \mid \mathbb{Q})=S_{n}$ for $E$ the splitting field over $\mathbb{Q}$ of a polynomial $f(x) \in \mathbb{Q}[x]$ of degree $n$, then $f(x)$ must be irreducible.
(b) If $\alpha$ is algebraic over $\mathbb{Q}$, then its minimum polynomial $\mu_{\alpha \mid \mathbb{Q}}(x)$ must be irreducible.
(c) If $A \in \mathbb{M}_{n \times n}(\mathbb{R})$, then its minimum polynomial $\mu_{A}(x)$ must be irreducible.
28. (Jan 12 # 8) Let $F$ be a field of characteristic zero, let $K$ and $L$ be finite extensions of $F$ and $K L$ the compositum of $K$ and $L$.
(a) Prove that $[K L: F] \leq[K: F][L: F]$.
(b) Assume that $[K: F]$ and $[L: F]$ are relatively prime. Prove that $[K L: F]=[K: F][L$ : $F]$.
(c) Give an example where $K \cap L=F$ but $[K L: F] \neq[K: F][L: F]$.
(d) Assume that $K / F$ and $L / F$ are both Galois. Prove that $\operatorname{Gal}(K L / F)$ is isomorphic to a subgroup of $\operatorname{Gal}(K / F) \times \operatorname{Gal}(L / F)$. (You need not prove that $K L / F$ is Galois).
Note: The assertions of (a),(b) and (d) remain valid for $F$ of positive characteristic, but part (a) has a shorter proof in the case of characteristic zero.
29. (Aug 12 #7) Let $K / F$ be a field extension, let $\alpha, \beta \in K \backslash F$ be algebraic over $F$, and let $p=\operatorname{deg}_{F}(\alpha)$ and $q=\operatorname{deg}_{F}(\beta)$. Suppose that $p$ and $q$ are distinct primes and that $p>q$.
(a) Prove that $[F(\alpha, \beta): F]=p q$.
(b) Prove that $\operatorname{deg}_{F}(\alpha \beta)=p$ or $p q$.
(c) Give an example showing that it MAY happen that $\operatorname{deg}_{F}(\alpha \beta)=p$.
30. (Jan $13 \# 4$ ) Let $p$ be a prime, $\mathbb{F}_{p}$ a finite field of order $p$, and let $F$ be a fixed algebraic closure of $\mathbb{F}_{p}$. For $n \in \mathbb{N}$, denote by $\mathbb{F}_{p^{n}}$ the unique subfield of order $p^{n}$ inside $F$.
(a) Prove that $\mathbb{F}_{p^{n}} \cup \mathbb{F}_{p^{m}}$ is a subfield if and only if $m$ divides $n$ or $n$ divides $m$.
(b) For a subset $S$ of $\mathbb{N}$, let

$$F(S)=\bigcup_{n \in S} \mathbb{F}_{p^{n}}$$

Give an example (with proof) of an infinite set $S$ for which $F(S)$ is a subfield and $F(S) \neq F$.
31. (Aug $13 \# 6$ ) Let $p$ be a prime and $\zeta$ a primitive $p^{t h}$ root of unity (in $\mathbb{C}$ ). Set $R:=\mathbb{Z}[\zeta]$ and $K:=\mathbb{Q}(\zeta)$.
(a) Show that $R$ is a free $\mathbb{Z}$-module and $R \cap \mathbb{Q}=\mathbb{Z}$.
(b) Identify $\operatorname{Gal}(K \mid \mathbb{Q})$ and show that the natural action of $\operatorname{Gal}(K \mid \mathbb{Q})$ on $K$ sends elements of $R$ to itself (hence giving an action of $\operatorname{Gal}(K \mid \mathbb{Q})$ on $R$ ).
(c) For any two integers $m, n$ which are not divisible by $p$, show that the quotient ( $1- \left.\zeta^{m}\right) /\left(1-\zeta^{n}\right)$ is an element of $R$.
Hint: Reduce to the case where $n$ divides $m$.
(d) Verify that $p=(1-\zeta) \ldots\left(1-\zeta^{p-1}\right)$.

Hint: manipulate the cyclotomic polynomial associated to $\zeta$.
(e) Prove that $1-\zeta$ is not a unit of $R$.
(f) Prove (using norms) that $1-\zeta$ is an irreducible element of $R$. (It is true, but harder to prove, that $1-\zeta$ is in fact a prime element of $R$.)
32. (Jan $14 \# 6$ ) Let $K \mid F$ be an extension of finite fields, and let $L, M$ be subfields of $K$ containing $F$. Assume that $L \cap M=F$.
(a) Show that the degrees $[L: F]$ and $[M: F]$ are relatively prime.

Hint: How many subfields of a given order does a finite field have?
(b) Now assume additionally that $L=F(\alpha), M=F(\beta)$ and $K=F(\alpha, \beta)$ for some $\alpha, \beta \in K$. Prove that $K=F(\alpha+\beta)$.
33. (Aug 18 #8) Describe the splitting field of $x^{4}+x^{2}+1$ over $\mathbb{Q}$. That is, describe elements of the algebraic closure of $\mathbb{Q}$ which need to be adjoined in order to obtain the splitting field. What is the degree of the splitting field over $\mathbb{Q}$ ?
34. (Jan $19 \# 8$ ) Let $p>2$ be a prime, let $\mathbb{F}_{p}$ be a field of order $p$ and let $\overline{\mathbb{F}_{p}}$ be an algebraic closure of $\mathbb{F}_{p}$. Let $f(x)=x^{m}+1$ for some $m \in \mathbb{N}$. Assume that $f$ is irreducible in $\mathbb{F}_{p}[x]$, and let $\alpha$ be a root of $f$ in $\overline{\mathbb{F}_{p}}$.
(a) Prove that the multiplicative order of $\alpha$ is equal to $2 m$.
(b) Prove that $2 m$ divides $p^{m}-1$ and $2 m$ does not divide $p^{k}-1$ for any $0<k<m$.
(c) Prove that $m \neq 4$.
35. (Aug 19 #8) Let $q$ be a prime power, let $\mathbb{F}_{q}$ be a field of order $q$, and let $a \in \mathbb{F}_{q}$ be a nonzero element. Consider the polynomial

$$f(x)=x^{q+1}-a$$

Let $L$ be the splitting field of $f(x)$ over $K=\mathbb{F}_{q}$. Prove that $[L: K]=2$. Note: Make sure to prove that the degree is equal to 2 , not just $\leq 2$.
Hint: Let $\alpha$ be a root of $f$. What can you say about the (multiplicative) order of $\alpha$ ?
36. (Aug $20 \# 7$ ) Let $p$ be an odd prime number and set $q=p^{2}$. Consider the finite field $\mathbb{F}_{q}$.
(a) Show that $\mathbb{F}_{q}$ contains a primitive 8-th root of unity $w$ and that the element $\alpha=w+w^{-1}$ satisfies $\alpha^{2}=2$.
(b) Show that $\alpha \in \mathbb{F}_{p}$ if and only if $p \equiv 1$ or $7 \bmod 8$.

Hint: Check when is $\alpha^{p}=\alpha$.
37. (Jan 21 #7) Recall that a field $F$ is called perfect if either $F$ has characteristic zero or $F$ has characteristic $p>0$ and every element of $F$ is equal to $a^{p}$ for some $a \in F$. Prove that $F$ admits a finite inseparable extension if and only if $F$ is not perfect.
38. (Aug 21 #7)
(a) Let $n \in \mathbb{N}$, let $F$ be an algebraically closed field either of characteristic 0 or of characteristic $p>0$ where $p$ does not divide $n$. Prove that $F$ contains a primitive $n^{\text {th }}$ root of unity.
(b) According to (a) $\overline{\mathbb{F}}_{3}$ (the algebraic closure of a field with 3 elements) contains a primitive $13^{\text {th }}$ root of unity, call it $\omega$. Compute $\left[\mathbb{F}_{3}(\omega): \mathbb{F}_{3}\right]$ (with proof).
39. (Jan $22 \# 7$ ) Let $p$ be a prime and let $F$ be a field of order $p^{6}$. Let

$$\operatorname{Prim}(F)=\left\{\alpha \in F: \mathbb{F}_{p}(\alpha)=F\right\} .$$

(a) Find (with proof) an explicit formula for $|\operatorname{Prim}(F)|$. Hint: It is easier to compute $|F \backslash \operatorname{Prim}(F)|$, the cardinality of the complement of $\operatorname{Prim}(F)$.
(b) Let $\operatorname{Irr}_{6}(p)$ denote the set of all monic irreducible polynomials of degree 6 over $\mathbb{F}_{p}$. Find (with proof) a simple relation between $\left|\operatorname{Irr}_{6}(p)\right|$ and $|\operatorname{Prim}(F)|$.