Math Problems Vault

Problem 67.1

Let $A$ be an algebra over a field $K$ and

K=K_{0} \subset K_{1} \subset K_{1} \subset \ldots \subset K_{s}

a tower of subfields in $A$ .

Prove that

(A: K)=\left(A: K_{s}\right)\left(K_{s}: K_{s-1}\right) \ldots \left(K_{1}: K_{0}\right)

?

Problem 67.2

(a)(b)(c)(d)

Let $A$ be an algebra over a field $K$ and $a \in A$ .

Prove that:

?

(a)

if the element $a$ is not algebraic over $K$ , then the subalgebra $K[a]$ is isomorphic to the ring of polynomials $K[x]$ ;

(b)

if $a$ is an algebraic element over $K$ , then

K[a] \simeq K[x] / \left\langle \mu _{a}(x)\right\rangle ,

where $\mu_{a}(x)$ is a certain uniquely determined monic polynomial (the minimal polynomial of the element $a$ ) over $K$ ;

(c)

if $A$ is a field, then for every element $a \in A$ algebraic over $K$ , the polynomial $\mu_{a}(x)$ is irreducible in $K[x]$ ;

(d)

if every element of $A$ is algebraic over $K$ and, for every $a \in A$ , the polynomial $\mu_{a}(x)$ is irreducible, then $A$ is a field.

Problem 67.3

Find the minimal polynomials of the elements:

?

(a)

$\sqrt{2}$ over $\mathbb {Q}$ ;

(b)

$\sqrt[7]{5}$ over $\mathbb {Q}$ ;

(c)

$\sqrt[105]{9}$ over $\mathbb {Q}$ ;

(d)

$2-3 i$ over $\mathbb {R}$ ;

(e)

$2-3 i$ over $\mathbb {C}$ ;

(f)

$\sqrt{2}+\sqrt{3}$ over $\mathbb {Q}$ ;

(g)

$1+\sqrt{2}$ over $\mathbb {Q}(\sqrt{2}+\sqrt{3})$ .

Problem 67.4

(a)(b)

Prove that:

?

(a)

if $A$ is a finite-dimensional algebra over $K$ , then every element of $A$ is algebraic over $K$ ;

(b)

if $a_{1}, \ldots , a_{s} \in A$ are algebraic elements over $K$ , then the subalgebra $K\left[a_{1}, \ldots , a_{s}\right]$ is finite-dimensional over $K$ .

Problem 67.5

Prove that if $A$ is a field and $a_{1}, \ldots , a_{s} \in A$ are algebraic elements over $K$ , then the extension $K\left(a_{1}, \ldots , a_{s}\right)$ coincides with the algebra $K\left[a_{1}, \ldots , a_{s}\right]$ .

?

Problem 67.6

Prove that the set of all elements of a $K$ -algebra $A$ algebraic over $K$ is a subalgebra of $A$ , and, if $A$ is a field, a subfield.

?

Problem 67.7

Prove that if, in a tower of fields

K=K_{0} \subset K_{1} \subset K_{2} \subset \ldots \subset K_{s}=L

each step $K_{i-1} \subset K_{i}(i=1, \ldots , s)$ is an algebraic extension, then $L / K$ is an algebraic extension.

?

Problem 67.8

Prove that every polynomial with coefficients in a field $K$ has a root in some extension $L / K$ .

?

Problem 67.9

(a)(b)

Let $K$ be a field. Prove that:

?

(a)

for an arbitrary polynomial from $K[x]$ there exists a splitting field of this polynomial over $K$ ;

(b)

for any finite set of polynomials from $K[x]$ there exists a splitting field over $K$ .

Problem 67.10

Let $K$ be a field, $g(x) \in K[x], h(x) \in K[x], f(x)=g(h(x))$ , and let $\alpha$ be a root of the polynomial $g(x)$ in some extension $L / K$ . Prove that the polynomial $f$ is irreducible over $K$ if and only if $g(x)$ is irreducible over $K$ and $h(x)-\alpha$ is irreducible over $K[\alpha ]$ .

?

Problem 67.11

(a)(b)(c)

Let $K$ be a field, $a \in K$ . Prove that:

?

(a)

if $p$ is a prime number, then the polynomial $x^{p}-a$ is either irreducible or has a root in $K$ ;

(b)

if the polynomial $x^{n}-1$ splits into linear factors in $K[x]$ , then the polynomial $x^{n}-a \in K[x]$ is either irreducible, or for some divisor $d \neq 1$ of $n$ the polynomial $x^{d}-a$ has a root in $K$ ;

(c)

the assumption that $x^{n}-1$ splits into linear factors is essential for statement (b) to hold.

Problem 67.12

Prove that over a field $K$ of characteristic $p \neq 0$ , the polynomial $f(x)=x^{p}-x-a$ is either irreducible or splits into a product of linear factors, and give this factorization if $f(x)$ has a root $x_{0}$ .

?

Problem 67.13

Find the degree of the splitting field over $\mathbb {Q}$ for the polynomials:

?

(a)

$a x+b \quad (a, b \in \mathbb {Q}, \quad a \neq 0)$ ;

(b)

$x^{2}-2 ; \quad$

(c)

$x^{3}-1 ; \quad$

(d)

$x^{3}-2 ; \quad$

(e)

$x^{4}-2$ ;

(f)

$x^{p}-1 \quad (p$ a prime number);

(g)

$x^{n}-1 \quad (n \in \mathbb {N})$ ;

(h)

$x^{p}-a \quad (a \in \mathbb {Q}$ and not a $p$ -th power in $\mathbb {Q}, p$ a prime number);

(i)

$\left(x^{2}-a_{1}\right) \ldots \left(x^{2}-a_{n}\right) \quad (a_{1}, \ldots , a_{n}$ belong to $\mathbb {Q}^{*}$ and are all distinct).

Problem 67.14

Prove that a finite extension $L / K$ is simple if and only if the set of intermediate fields between $K$ and $L$ is finite, and give an example of a finite extension that is not simple.

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Problem 67.15

Let $L / K$ be an algebraic extension. Prove that the extension $L(x) / K(x)$ is also algebraic and

(L(x): K(x))=(L: K)

?

Problem 67.16

Let $L / K$ be an extension. Elements $a_{1}, \ldots , a_{s} \in L$ are called algebraically independent over $K$ if $f\left(a_{1}, \ldots , a_{s}\right) \neq 0$ for every nonzero polynomial $f\left(x_{1}, \ldots , x_{s}\right) \in K\left[x_{1}, \ldots , x_{s}\right]$ .

Prove that the elements $a_{1}, \ldots , a_{s} \in L$ are algebraically independent over $K$ if and only if the extension $K\left(a_{1}, \ldots , a_{s}\right)$ is $K$ -isomorphic to the field of rational functions $K\left(x_{1}, \ldots , x_{s}\right)$ .

?

Problem 67.17

Let $L / K$ be an extension and let $a_{1}, \ldots , a_{s} ; b_{1}, \ldots , b_{n}$ be two maximal systems of elements of $L$ algebraically independent over $K$ . Prove that $m=n$ (the transcendence degree of $L$ over $K$ ).

?

Problem 67.18

Prove that:

?

(a)

a finite-dimensional commutative $K$ -algebra $A$ has only finitely many maximal ideals, and their intersection coincides with the set $N(A)$ of all nilpotent elements of the algebra $A$ (the nilradical of the algebra $A$ );

(b)

$A^{\mathrm{red}}=A / N(A)$ is a reduced algebra (contains no nonzero nilpotent elements);

(c)

the algebra $A / N(A)$ is isomorphic to the direct product of fields $K_{1}, \ldots , K_{s}$ , which are extensions of the field $K$ ;

(d)

$s \leqslant (A: K)$ ;

(e)

the set of extensions $K_{i}$ is determined for the algebra $A$ uniquely up to isomorphism¹;

The extensions $K_{1}, \ldots , K_{s}$ , together with the canonical homomorphisms $A \rightarrow K_{s}$ , are called the components of the algebra $A$ ↩

(f)

if $B$ is a subalgebra of $A$ , then every component of $B$ is an extension embedded in one or more components of $A$ ;

(g)

if $I$ is an ideal of $A$ , then the components of the algebra $A / I$ are among the components of the algebra $A$ .

Problem 67.19

Let $K$ be a field, $f(x) \in K[x]$ , $p_{1}(x)^{k_{1}} \ldots p_{s}(x)^{k_{s}}$ the factorization of $f(x)$ into a product of powers of distinct irreducible polynomials over $K$ , $A=K[x] /\langle f(x)\rangle$ . Prove that

A^{\mathrm{red}}=A / N(A) \simeq \prod _{i=1}^{s} k[x] / \left\langle p_{l}(x)\right\rangle

?

Problem 67.20

(a)(b)

Let $A$ be a $K$ -algebra and $L$ an extension of the field $K$ .

Prove that:

?

(a)

if $f_{1}, \ldots f_{n}$ are distinct $K$ -homomorphisms $A \rightarrow L$ , then $f_{1}, \ldots , f_{n}$ are linearly independent as elements of the vector space over $L$ of all $K$ -linear maps $A \rightarrow L$ ;

(b)

the number of distinct $K$ -homomorphisms $A \rightarrow L$ does not exceed $(A: L)$ .

Find all automorphisms of the fields $\mathbb {Q}(\sqrt{2}), \mathbb {Q}(\sqrt{2}+\sqrt{3}), \mathbb {Q}(\sqrt[3]{2})$ .

Problem 67.21

(a)(b)

Let $A$ be a finite-dimensional $K$ -algebra and $L$ an extension of the field $K$ . Set $A_{L}=L \otimes_{K} A$ . Let $\left(e_{1}, \ldots , e_{n}\right)$ be a basis of $A$ over $K$ . Prove that:

?

(a)

$\left(1 \otimes e_{1}, \ldots , 1 \otimes e_{n}\right)$ is a basis of $A_{L}$ over $L$ ;

(b)

under the natural embedding of $A$ in $A_{L}$ , the image of $A$ is a $K$ -subalgebra of $A_{L}$ .

Problem 67.22

Let $A$ be a finite-dimensional $K$ -algebra, $L$ an extension of the field $K$ . Prove that:

?

(a)

if $B$ is a subalgebra of $A$ , then $B_{L}$ is a subalgebra of $A_{L}$ ;

(b)

if $I$ is an ideal of the algebra $A$ and $I_{L}$ the corresponding ideal of $A_{L}$ , then $(A / I)_{L} \simeq A_{L} / I_{L}$ ;

(c)

if $A=\prod_{i=1}^{s} A_{i}$ , then $A_{L} \simeq \prod_{i=1}^{s}\left(A_{i}\right)_{L}$ ;

(d)

if $K_{1}, \ldots , K_{s}$ is the set of components of the algebra $A$ , then the set of components of the algebra $A_{L}$ coincides with the union of the sets of components of the algebras $\left(K_{1}\right)_{L}, \ldots ,\left(K_{s}\right)_{L}$ ;

(e)

if $F$ is an extension of the field $L$ , then $\left(A_{L}\right)_{F} \cong A_{F}$ .

Problem 67.23

(a)(b)(c)

Let $A$ be a finite-dimensional $K$ -algebra, $L$ an extension of the field $K$ , $B$ some $L$ -algebra. Prove that:

?

(a)

every $K$ -homomorphism $A \rightarrow B$ extends uniquely to an $L$ -homomorphism $A_{L} \rightarrow B_{L}$ ;

(b)

the set of $K$ -homomorphisms $A \rightarrow L$ is in bijective correspondence with the set of components of the algebra $A_{L}$ isomorphic to $L$ ;

(c)

the number of distinct $K$ -homomorphisms $A \rightarrow L$ does not exceed $(A: K)$ (cf. problem 67.20, (b)).

Problem 67.24

Let $F$ and $L$ be extensions of the field $K$ , with $F$ finite. Prove that there exists an extension $E / K$ for which there are embeddings of $F$ into $E$ and of $L$ into $E$ that fix every element of $K$ .

?

Problem 67.25

(a)(b)

Let $A$ be a finite-dimensional $K$ -algebra and $A=K\left[a_{1}, \ldots , a_{s}\right]$ . Prove that the following properties of an extension $L / K$ are equivalent:

?

(a)

all components of $A_{L}$ are isomorphic to $L$ ;

(b)

$L$ is a splitting field for the minimal polynomial of every element $a \in A$ ( $L$ is called a splitting field of the $K$ -algebra $A$ ).

Problem 67.26

Prove that if $L$ is a splitting field of the $K$ -algebra $A$ and $B$ is a subalgebra of $A$ , then any $K$ -homomorphism $B \rightarrow L$ extends to a $K$ -homomorphism $A \rightarrow L$ .

?

Problem 67.27

(a)(b)(c)

A splitting field $L$ for a finite-dimensional $K$ -algebra $A$ is called a decomposition field for $A$ if no proper subfield of it containing $K$ is a splitting field for $A$ . Prove that:

?

(a)

if $A=K\left[a_{1}, \ldots , a_{s}\right]$ , then $L$ is a decomposition field for $A$ if and only if $L$ is a splitting field for the minimal polynomials of the elements $a_{1}, \ldots , a_{s}$ ;

(b)

any two decomposition fields of a $K$ -algebra $A$ are isomorphic over $K$ ;

(c)

for a decomposition field of a $K$ -algebra $A$ there exists a $K$ -embedding into any splitting field for $A$ .

Problem 67.28

Let $A$ be a finite-dimensional $K$ -algebra, $L$ a decomposition field for $A$ . Prove that the number of components of the $L$ -algebra $A_{L}$ is the same for all splitting fields of the algebra $A$ (the separable degree $(A: K)_{s}$ of the algebra $A$ ).

?

Problem 67.29

(a)(b)

Let $A$ be a $K$ -algebra and $L$ an extension of the field $K$ . Prove that:

?

(a)

the number of components of the algebra $A_{L}$ does not exceed $(A: K)_{s}$ ;

(b)

the number of distinct $K$ -homomorphisms $A \rightarrow L$ does not exceed $(A: K)_{s}$ , with equality if and only if $L$ is a splitting field for $A$ .

Problem 67.30

Prove that the following properties of a finite extension $L / K$ are equivalent:

?

(a)

all components of the algebra $L_{L}$ are isomorphic to $L$ ;

(b)

$L$ has $(L: K)$ $K$ -automorphisms;

(c)

for any $K$ -embeddings $\varphi_{i}: L \rightarrow L^{\prime }(i=1,2)$ of the field $L$ into any extension $L^{\prime } / K$ we have $\varphi_{1}(L)=\varphi_{2}(L)$ ;

(d)

every irreducible polynomial from $K[x]$ that has a root in $L$ splits over $L$ into a product of linear factors;

(e)

$L$ is the splitting field of some polynomial from $K[x]$ . (An extension $L / K$ satisfying these conditions is called normal.)

Problem 67.31

(a)(b)(c)

Let $K \subset L \subset F$ be a tower of finite extensions of a field $K$ .

Prove that:

?

(a)

if the extension $F / K$ is normal, then the extension $F / L$ is also normal;

(b)

if the extensions $L / K$ and $F / L$ are normal, the extension $F / K$ need not be normal;

(c)

every extension of degree 2 is normal.

Problem 67.32

(a)(b)(c)

Let $A$ be a finite-dimensional $K$ -algebra and $a \in A$ . The characteristic polynomial, determinant, and trace of the linear operator $t \mapsto a t$ on $A$ are denoted respectively by

\chi _{A / K}(a, x), \quad N_{A / K}(a), \quad \operatorname {tr}_{A / K}(a)

and are called, respectively, the characteristic polynomial, the norm, and the trace of the element $a$ of the algebra $A$ over $K$ .

Prove that if $K \subset L \subset F$ is a tower of finite field extensions and $a \in F$ , then:

?

(a)

$\chi_{F / K}(a, x)=N_{L(x) / K(x)}\left(\chi_{F / L}(a, x)\right), \quad$ where $\chi_{F / L}(a, x)$ is regarded as an element of the field of rational functions $K(x)$ ;

(b)

$N_{F / K}(a)=N_{L / K}\left(N_{F / L}(a)\right)$ ;

(c)

$\operatorname {tr}_{F / K}(a)=\operatorname {tr}_{L / K}\left(\operatorname {tr}_{F / L}(a)\right)$ .

Problem 67.33

(a)(b)

Let $L / K$ be a finite extension and $a \in L$ .

Prove that:

?

(a)

the minimal polynomial of the element $a$ equals $\pm \chi_{K(a) / K}(a, x)$ ;

(b)

$\chi_{L / K}(a, x)$ is (up to sign) a power of the minimal polynomial of the element $a$ .

Problem 67.34

Let $L / K$ be a finite extension. Prove that the $K$ -bilinear form on $L$

(x, y) \mapsto \operatorname {tr}_{L / K}(x y)

is either nondegenerate, or $\operatorname {tr}_{L / K}(x)=0$ for all $x \in L$ .

?

Problem 67.35

(a)(b)(c)(d)

Prove that the following properties of a finite-dimensional $K$ -algebra $A$ are equivalent:

?

(a)

for every extension $L / K$ the algebra $A_{L}$ is reduced (problem 67.18);

(b)

$(A: K)_{s}=(A: K)$ (problem 67.28);

(c)

for some extension $L / K$ there exist $(A: K)$ homomorphisms of $K$ -algebras $A \rightarrow L$ ;

(d)

the bilinear form $(x, y) \mapsto \operatorname {tr}_{A / K}(x y)$ on $A$ is nondegenerate. (An algebra $A$ satisfying these conditions is called separable.)

Problem 67.36

Let $L$ be an extension of a field $K$ . Prove that a finite-dimensional $K$ -algebra $A$ is separable if and only if the $L$ -algebra $A_{L}$ is separable.

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Problem 67.37

Prove that every subalgebra and every quotient algebra of a separable $K$ -algebra are separable $K$ -algebras.

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Problem 67.38

Let $A$ be a separable $K$ -algebra, $(A: K)=n, \varphi_{1}, \ldots , \varphi_{n}$ distinct $K$ -homomorphisms of the algebra $A$ into some splitting field $L$ of it. Prove that for every element $a \in A$

\begin{gathered} \operatorname {tr}_{A / K}(a)=\sum _{i=1}^{n} \varphi _{i}(a), \quad N_{A / K}(a)=\prod _{i=1}^{n} \varphi _{i}(a), \\ \chi _{A / K}(a, x)=\prod _{i=1}^{n}\left(\varphi _{i}(a)-x\right). \end{gathered}

?

Problem 67.39

(a)(b)

A finite extension $L / K$ is called separable if $L$ is a separable $K$ -algebra.

?

(a)

Prove that a separable field extension is simple.

(b)

Are the numbers

a=-\frac{1}{2}+i \frac{\sqrt{2}}{2}, \quad b=\sqrt{2}+i

primitive elements of the extension $\mathbb {Q}(\sqrt{2}, i) / \mathbb {Q}$ ?

Problem 67.40

Prove that a finite-dimensional $K$ -algebra is separable if and only if it is a direct product of separable extensions of the field $K$ .

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Problem 67.41

Let $K=K_{0} \subset K_{1} \subset \ldots \subset K_{s}=L$ be a tower of finite field extensions. Prove that the extension $L / K$ is separable if and only if every extension $K_{i} / K_{i-1}(i=1, \ldots , s)$ is separable.

?

Problem 67.42

(a)(b)

Let $K$ be a field. A polynomial $f(x) \in K[x]$ is called separable if it has no multiple roots in any extension of the field $K$ .

Prove that:

?

(a)

if $K$ has characteristic 0, then every irreducible polynomial from $K[x]$ is separable;

(b)

if $K$ has characteristic $p \neq 0$ , then an irreducible polynomial $f(x) \in K[x]$ is separable if and only if it cannot be written in the form $g\left(x^{p}\right)$ , where $g(x) \in K[x]$ .

Give an example of an inseparable irreducible polynomial over some field.

Problem 67.43

Let $A$ be a finite-dimensional $K$ -algebra. An element $a \in A$ is called separable over the field $K$ if $K[a]$ is a separable $K$ -algebra. Prove that an element is separable if and only if its minimal polynomial is separable.

?

Problem 67.44

(a)(b)

Let $K \subset L \subset F$ be a tower of finite field extensions.

Prove that:

?

(a)

if an element $a \in F$ is separable over $K$ , then $a$ is separable over $L$ ;

(b)

the converse of (a) holds if the extension $L / K$ is separable.

Problem 67.45

Let $A$ be a separable $K$ -algebra, $f(x) \in K[x]$ a separable polynomial.

Prove that the algebra $B=A[x] /\langle f(x)\rangle$ is separable.

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Problem 67.46

(a)(b)(c)

Let $A=K\left[a_{1}, \ldots , a_{s}\right]$ be a finite-dimensional $K$ -algebra. Prove that the following conditions are equivalent:

?

(a)

$A$ is a separable $K$ -algebra;

(b)

every element $a \in A$ is separable;

(c)

the elements $a_{1}, \ldots , a_{s}$ are separable.

Problem 67.47

(a)(b)

Prove that:

?

(a)

a finite extension $K / F$ of a field $F$ is separable if and only if either $K$ has characteristic 0, or the characteristic of $K$ equals $p>0$ and $K^{p}=K$ ;

(b)

every finite extension of a finite field is separable.

Problem 67.48

A finite field extension $L / K$ of characteristic $p>0$ is called purely inseparable if there are no separable elements over $K$ in $L \backslash K$ . Prove that $L / K$ is a purely inseparable extension if and only if $L^{p^{k}} \subseteq K$ for some $k \geqslant 1$ .

?

Problem 67.49

Let $K \subset K_{0} \subset K_{1} \ldots \subset K_{s}=L$ be a tower of finite field extensions. Prove that the extension $L / K$ is purely inseparable if and only if every extension $K_{i} / K_{i-1}(i=1, \ldots , s)$ is purely inseparable.

?

Problem 67.50

Prove that the degree of a purely inseparable extension of a field of characteristic $p>0$ is a power of $p$ , and its separable degree equals 1.

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Problem 67.51

(a)(b)(c)(d)

Let $L / K$ be a finite field extension.

Prove that:

?

(a)

the set $K_{s}$ of all elements of $L$ separable over $K$ is a field, separable over $K$ ;

(b)

$L / K_{s}$ is a purely inseparable extension;

(c)

$\left(K_{s}: K\right)=(L: K)_{s}$ ;

(d)

$(L: K)=(L: K)_{s} \cdot (L: K)_{i}$ , where $(L: K)_{i}=\left(L: K_{s}\right)$ is the inseparable degree of the extension $L / K$ .

Problem 67.52

(a)(b)

Let $K \subset L \subset F$ be a tower of finite field extensions. Prove that:

?

(a)

$(F: K)_{s}=(F: L)_{s} \cdot (L: K)_{s}$ ;

(b)

$(F: K)_{i}=(F: L)_{i} \cdot (L: K)_{i}$ .

Problem 67.53

(a)(b)(c)

Let $L / K$ be a finite field extension, $n=(L: K)_{s}$ , and let $\varphi_{1}, \ldots , \varphi_{n}$ be the set of all $K$ -embeddings of the field $L$ into some splitting field of the extension $L / K$ .

Prove that for any $a \in L$ :

?

(a)

$\operatorname {tr}_{L / K}(a)=(L: K)_{i} \sum_{j=1}^{n} \varphi_{j}(a)$ ;

(b)

$N_{L / K}(a)=\left(\prod_{j=1}^{n} \varphi_{j}(a)\right)^{(L: K)_{i}}$ ;

(c)

$\chi_{L / K}(a, x)=\left(\prod_{j=1}^{n}\left(\varphi_{j}(a)-x\right)\right)^{(L: K)_{i}}$ .

Problem 67.54

(a)(b)

A normal finite separable field extension $L / K$ is called a Galois extension, and the group of $K$ -automorphisms of such an extension is called its Galois group, denoted by $G(L / K)$ .

Prove that:

?

(a)

$G(L / K)$ acts transitively on the set of roots, lying in $L$ , of the minimal polynomial of any element of $L$ ;

(b)

the order of the group $G(L / K)$ equals the degree of the extension $L / K$ .

Problem 67.55

(a)(b)(c)(d)

Find the Galois group of the extension:

?

(a)

$\mathbb {C} / \mathbb {R} ; \quad$

(b)

$\mathbb {Q}(\sqrt{2}) / \mathbb {Q}$ ;

(c)

$L / K$ , where $(L: K)=2$ ;

(d)

$\mathbb {Q}(\sqrt{2}+\sqrt{3}) / \mathbb {Q}$ .

Problem 67.56

The Galois group over a field $K$ of a separable polynomial $f(x) \in K[x]$ is the Galois group of the splitting field of this polynomial over $K$ (as a certain permutation group on the set of roots of $f(x)$ ). Find the Galois groups over the field $\mathbb {Q}$ of the polynomials:

?

(a)

$a x+b \quad (a, b \in \mathbb {Q}, \quad a \neq 0)$ ;

(b)

$x^{2}-2$ ;

(c)

$x^{3}-1$ ;

(d)

$x^{3}-2$ ;

(e)

$x^{4}-2$ ;

(f)

$x^{p}-1 \quad (p$ a prime number);

(g)

$x^{n}-1 \quad (n \in \mathbb {N})$ ;

(h)

$x^{p}-a \quad (a \in \mathbb {Q}$ and not a $p$ -th power in $\mathbb {Q}, p$ a prime number);

(i)

$\left(x^{2}-a_{1}\right) \ldots \left(x^{2}-a_{n}\right) \quad (a_{1}, \ldots , a_{n}$ belong to $\mathbb {Q}^{*}$ and are all distinct).

Problem 67.57

Let $G$ be a finite group of automorphisms of a field $L$ and $K=L^{G}$ the field of fixed elements. Prove that $L / K$ is a Galois extension and $G(L / K)=G$ .

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Problem 67.58

Prove that if the elements $a_{1}, \ldots , a_{n}$ are algebraically independent over a field $K$ , then the Galois group of the polynomial

x^{n}+a_{1} x^{n-1}+\ldots +a_{n}

over the field of rational functions $K\left(a_{1}, \ldots , a_{n}\right)$ is $\mathbf{S}_{n}$ .

?

Problem 67.59

Prove that every finite group is the Galois group of some field extension.

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Problem 67.60

Let $L / K$ be a Galois extension and $G$ its Galois group. Prove that assigning to every subgroup $H \subset G$ the subfield $L^{H}$ of fixed elements defines a bijective correspondence between all subgroups of the group $G$ and all intermediate subfields of the extension $L / K$ , under which the intermediate subfield $F$ corresponds to the subgroup $H=G(L / F)$ ; moreover, the extension $F / K$ is normal if and only if the subgroup $H$ is normal in $G$ , and in that case the canonical map $G \rightarrow G(F / K)$ defines an isomorphism $G(F / K) \simeq G / H$ .

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Problem 67.61

Using the fundamental theorem of Galois theory and the existence of a real root for every polynomial of odd degree with real coefficients, prove the algebraic closedness of the field of complex numbers.

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Problem 67.62

Prove that the Galois group of every finite extension $L / \mathbb {F}_{p}$ is cyclic and is generated by the automorphism $x \mapsto x^{p}(x \in L)$ .

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Problem 67.63

Prove that the Galois group over a field $K$ of a separable polynomial $f(x) \in K[x]$ , regarded as a subgroup of $\mathbf{S}_{n}$ , is contained in the group of even permutations if and only if the discriminant

D=\prod _{i>j}\left(x_{i}-x_{j}\right)^{2}

of the polynomial $f(x)$ , where $x_{1}, \ldots , x_{n}$ are the roots of $f(x)$ in its splitting field, is a square in the field $K$ .

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Problem 67.64

Let $L / K$ be a Galois extension with cyclic Galois group $\langle \varphi \rangle_{n}$ . Prove that there exists an element $a \in L$ such that the elements $a, \varphi (a), \ldots , \varphi^{n-1}(a)$ form a basis of $L$ over $K$ .

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Problem 67.65

Let $L / K$ be a separable extension of degree $n$ and let $\varphi_{1}, \ldots , \varphi_{n}$ be the distinct $K$ -embeddings of $L$ into some field splitting for $L$ . Prove that an element $a \in L$ is a primitive element of $L / K$ if and only if the images $\varphi_{1}(a), \ldots , \varphi_{n}(a)$ are distinct.

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Problem 67.66

Find the automorphism group of the $K$ -algebra that is the direct product of $n$ fields isomorphic to $K$ .

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Problem 67.67

Let $L / K$ be a Galois extension with Galois group $G$ , $L=\prod L_{\sigma }$ , where $L_{\sigma }$ is the component of the algebra $L_{L}$ onto which the projection induces the automorphism $\sigma$ of $L$ , and $e_{\sigma }$ the unity of the component $L_{\sigma }$ . Prove that for the extensions of the automorphisms of $G$ to $L$ -automorphisms of the algebra $L_{L}$ , the equalities

\tau \left(e_{\sigma }\right)=e_{\sigma \tau ^{-1}}, \quad \sigma , \tau \in G

hold.

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Problem 67.68

Let $L$ be a splitting field for a separable $K$ -algebra $A$ and let $\varphi_{1}, \ldots , \varphi_{n}$ be the set of all $K$ -homomorphisms $A \rightarrow L$ . Prove that elements $y_{1}, \ldots , y_{n} \in A$ form a basis of $A$ over $K$ if and only if $\operatorname {det}\left(\varphi_{i}\left(y_{j}\right)\right) \neq 0$ .

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Problem 67.69

Prove that in a Galois extension $L / K$ with Galois group $G$ , there exists an element $a \in L$ such that the set $\left\{ \sigma (a) \mid \sigma \in G\right\}$ is a basis of the field $L$ over $K$ .

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Problem 67.70

Find the field of invariants $K\left(x_{1}, \ldots , x_{n}\right)^{A_{n}}$ for the group $\mathbf{A}_{n}$ , acting on the field of rational functions by permuting the variables.

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Problem 67.71

Let $\varepsilon$ be a primitive complex $n$ -th root of 1 and let the group $G=\langle \sigma \rangle_{n}$ act on the field $\mathbb {C}\left(x_{1}, \ldots , x_{n}\right)$ by the rule

\sigma \left(x_{1}\right)=\varepsilon ^{i} x_{i} \quad (i=1, \ldots , n)

Find the field of invariants $\mathbb {C}\left(x_{1}, \ldots , x_{n}\right)^{G}$ .

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Problem 67.72

Find the field of invariants for the group $G$ acting on the field $\mathbb {C}\left(x_{1}, \ldots , x_{n}\right)$ by cyclic permutation of the variables.

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Problem 67.73

Let the field $K$ contain all $n$ -th roots of 1, and let an element $a \in K$ not be a power with exponent $d>1$ for any divisor $d$ of $n$ . Find the Galois group over $K$ of the polynomial $x^{n}-a$ .

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Problem 67.74

Let the field $K$ contain all $n$ -th roots of 1 and let $L / K$ be a Galois extension with cyclic Galois group of order $n$ . Prove that $L=K(\sqrt[n]{a})$ for some element $a \in K$ .

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Problem 67.75

Let the field $K$ contain all $n$ -th roots of 1. Prove that a finite extension $L / K$ is a Galois extension with abelian Galois group of exponent $n$ if and only if

L=K\left(\theta _{1}, \ldots , \theta _{s}\right)

where

\theta _{i}^{n}=a_{i} \in K \quad (i=1, \ldots , s)

(i.e. $L$ is the splitting field over $K$ of the polynomial $\prod_{i=1}^{s}\left(x_{i}^{n}-a_{i}\right)^{s}$ ).

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Problem 67.76

Let the field $K$ contain all $n$ -th roots of 1 and let $L=K\left(\theta_{1}, \ldots , \theta_{s}\right)$ , where

\theta _{i}^{n}=a_{i} \in K^{*} \quad (i=1, \ldots , s).

Prove that

G(L / K) \simeq \left\langle \left(K^{*}\right)^{n}, a_{1}, \ldots , a_{s}\right\rangle /\left(K^{*}\right)^{n}

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Problem 67.77

Let the field $K$ contain all $n$ -th roots of 1. Establish a bijective correspondence between the set of all Galois extensions (up to $K$ -isomorphism) with abelian Galois group of exponent $n$ , and the set of all finite subgroups of the group $K^{*} /\left(K^{*}\right)^{n}$ .

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Problem 67.78

Prove that every Galois extension $L / K$ of degree $p$ of a field $K$ of characteristic $p>0$ has the form $L=K(\theta )$ , where $\theta$ is a root of the polynomial $x^{p}-x-a(a \in K)$ , and, conversely, every such extension is a Galois extension of degree 1 or $p$ .

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Problem 67.79

Let $K$ be a field of characteristic $p>0$ . Prove that a finite extension $L / K$ is a Galois extension of exponent $p$ if and only if $L=K\left(\theta_{1}, \ldots , \theta_{s}\right)$ , where $\theta_{i}$ is a root of the polynomial $x^{p}-x-a_{i}\left(a_{i} \in K ; i=1, \ldots , s\right)$ .

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Problem 67.80

Let $K$ be a field of characteristic $p>0$ and $L=K\left(\theta_{1}, \ldots , \theta_{s}\right)$ , where $\theta_{i}$ is a root of the polynomial $x^{p}-x-a_{i}\left(a_{i} \in K, i=1, \ldots , s\right)$ . Prove that

G(L / K) \simeq \left\langle \rho (K), a_{1}, \ldots , a_{s}\right\rangle / K

where $\rho : K \rightarrow K$ is the additive homomorphism $x \mapsto x^{p}-x$ .

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Problem 67.81

Let $K$ be a field of characteristic $p>0$ . Establish a bijective correspondence between the set of all Galois extensions (up to $K$ -isomorphism) $L / K$ with abelian Galois group of exponent $p$ , and the set of all finite subgroups of the group $K / \rho (K)$ .

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Footnotes