Tagged Questions

0
votes
0answers
29 views

The subset of a field has finite dimension

Let $K,E,$ and $F$ be fields such that $F \subset E \subset K$ and $[K:F]$ is finite . Show that $[E:F]$ is finite. I thought this was obvious since $[K:F]=[K:E][E:F]$ , then $[E:F]$ is definitely ...
3
votes
1answer
39 views

Why is $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] = p-1 $?

I know that $e^{2\pi i/p}$ is a root of $x^{p}-1$ and we can write: $ x^{p}-1=\left(x-1\right)\left(\sum_{i=0}^{p-1}x^{i}\right) $ So $e^{2\pi i/p}$ is a root of $\sum_{i=0}^{p-1}x^{i}$, which means ...
2
votes
0answers
37 views

Field extension

I need to prove that if $F$ is a field and $u=\frac{f(t)}{g(t)} \in F(t)$ (where $f,g$ are coprime in $F[t]$) then $[F(t):F(u)]=\max(\deg f,\deg g)$. I know I have to prove that $ug(x)-f(x)$ is ...
6
votes
1answer
44 views

Given $G$, when can we find a division ring $R$ with $R^*=G$?

This is motivated by a characterization of finite cyclic groups, in which one proves Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic. The proof is ...
4
votes
0answers
24 views

Is there a general algorithm to find a primitive element of a given finite extension (with a finite number of intermediate fields)?)

I just want to know if there is an algorithm to find the primitive element of a given finite extension $F/k$ if the intermediate fields are given. I know how to approach it in particular examples ...
1
vote
1answer
89 views

Calculating The Galois Group of the Splitting Field of $f=x^3-3$

If we let $f=x^3-3$ the let L be the splitting field for this polynomial I am trying to find $\Gamma(L:\mathbb{Q})$ and all intermediate field extensions. Now as this is a splitting field and finite ...
2
votes
1answer
22 views

Characteristic of commutative semisimple rings?

In one of my questions (Structure of the group ring of a direct product?), a statement is made for a commutative semisimple ring of characteristic $p^t, t\geq1$. Now I don't understand why there ...
1
vote
1answer
19 views

minimal polynomial of an element

I we have that $\alpha$ is the positive real root of $f=x^4-3$ then the splitting field of $f$ over $\mathbb{Q}$ is $\mathbb{Q}(\alpha,i)$ if I then want to find the degree of the extension of this ...
2
votes
1answer
80 views

A basic question on factorization

Is the following true? If not, can anyone add some reasonable assumptions to make it true? Statement. Let $E/F$ be a field extension and let $\alpha \in E \setminus F$ be algebraic over $F$. If $f(x) ...
1
vote
1answer
30 views

Field, Euclidean division question.

Let $K$ be a field and $f \in K[x]$. Show that if there is some $a \in K$ such that $f(a)=0$, then $x-a$ divides $f$. My friend told me to use Euclidean division by $x-a$. Also show that a ...
0
votes
0answers
34 views

Transitive action on the set of algebra homomorphisms.

Let $k$ be a field, and $K/k$ be a Galois extension. Suppose $K'/k$ be an extension with $K'$ is a finitely generated $k$-algebra. Then the Galois group $\textrm{Gal}(K/k)$ acts canonically on the set ...
1
vote
2answers
38 views

Question in Hungerford regarding field extensions

In Algebra by Hungerford, page 237 the sketch of proof for Theorem 1.10: Theorem 1.10: If $K$ is a field and $f\in K[x]$ polynomial of degree $n$, then there exists a simple extension field $F = ...
-1
votes
4answers
55 views

Allowing the zero element in a field to have an inverse

In the definition of a field one of the required properties is that every element other than zero has a multiplicative inverse. It's vague whether the zero is forced not to have an inverse or not, ...
4
votes
3answers
114 views

Quadratic subfield of cyclotomic field

Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
3
votes
1answer
64 views

Galois group of irreducible quartic with real coefficients

Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
3
votes
1answer
73 views

Irreducible polynomial over $\mathbb Z/7\mathbb Z$

Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
1
vote
2answers
52 views

Example where $[E:\mathbb{Q}]<|\mathrm{Aut}_{\mathbb{Q}}E|$

Let $F$ be an algebraic closure of $\mathbb{Q}$ and let $E\subset F$ be a splitting field over $\mathbb{Q}$ of the set $\{x^{2}+a|a\in\mathbb{Q}\}$ so that $E$ is algebraic and Galois over ...
0
votes
1answer
28 views

Smallest Galois extension

Let $F$ be a finite dimensional Galois extension of $K$ and let $E$ be an intermediate field. Show that there is a unique smallest field $L$ such that $E\subset L\subset F$ and $L$ is Galois over $K$. ...
2
votes
1answer
36 views

Showing that a field extension is Galois

Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
8
votes
1answer
95 views

Algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$

Is there a concrete description of the algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$?
3
votes
1answer
43 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
3
votes
0answers
55 views

Hilbert’s zeros theorem, an application. (The algebraic variation)

Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
0
votes
0answers
24 views

If $K/L$ is normal and $L/F$ is purely inseparable, then $K/F$ is normal [duplicate]

This is a problem in Morandi's Field and Galois Theory, on page 49: Let $F\subset L\subset K$ be field extensions such that $K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is ...
0
votes
0answers
23 views

An element of an extension field has a rational function with coefficients in the field that equals it

I stumbled across a statement without proof that I am having a hard time understanding or providing myself with intuition for. Any help would be most appreciated. The statement reads: Since $\xi$ is ...
2
votes
2answers
77 views

No rational solutions of a system of equations

Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that \begin{matrix} a^2b+2b^2c+2ac^2=0\\ a^2c+ab^2+2bc^2=0\\ a^3+2b^3+4c^3+12abc=3. \end{matrix} I'm able to show that this ...
1
vote
0answers
38 views

Showing $f(x)=g(x^{p^a})$ over field of (prime) characteristic $p>0$.

Let $f$ be a non-constant irreducible polynomial over a field $F$ of (prime) characteristic $p>0$. I need to prove that $f$ can be presented as: $f(x)=g(x^{p^a})$, where $g$ is irreducible over ...
5
votes
1answer
136 views

An exercise on cyclic extensions of Hungerford's book, Algebra.

I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following: Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let ...
2
votes
2answers
47 views

Special elements of fields extensions

I was wondering if there is a method to find all elements $w\in F(\alpha_1,\ldots,\alpha_n)$ such that $F(w)=F(\alpha_1,\ldots,\alpha_n)$, where $\alpha_1,\ldots,\alpha_n$ are algebraic over the field ...
3
votes
3answers
129 views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
1
vote
1answer
65 views

Characterization of transcendental elements in algebraic function fields

I would like to prove this equivalence: Let $F|K$ be an algebraic function field. Then $z \in F$ is transcendental over $K$ if and only if $[F:K(z)] < \infty$. (This statement is Remark 1.1.2 ...
1
vote
2answers
54 views

Maps compatible with the Frobenius

Let $F$ be a field. Fix a separable closure $F^{sep}$. Consider the monoid whose elements are maps of sets $F^{sep} \to F^{sep}$ which are equivariant with respect to the Galois action. These maps ...
3
votes
0answers
84 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
2
votes
1answer
82 views

Splitting field that isn't a Galois extension

I'm trying to find a counter-example to following statement: if $K$ is the splitting field of $g\in F[x],$ then the extension $K/F$ is Galois. I know the statement is true if $g$ is separable, ...
4
votes
0answers
74 views

Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
1
vote
1answer
29 views

Finitely generated integral domain and finitely generated $k$-algebra.

Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
2
votes
2answers
74 views

Proving that the inverse of an algebraic element is algebraic

Let $E$ be an extension of a field $F$. Suppose $c \in E$ is algebraic over $F$ where $c \neq 0$. I want to prove that $c^{-1}$ is also algebraic over $F$. I feel like I'm missing something obvious ...
0
votes
0answers
61 views

Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
2
votes
2answers
101 views

Roots of unity in $\mathbb{Q}(\zeta_p)$ for $p$ an odd prime

I am confused about a last step in a proof that the only roots of unity in $\mathbb{Q}(\zeta_p)$ are $\pm\zeta_p^j$, where $\zeta_p=e^{2 \pi i/p}$, $p$ is an odd prime and $1\leq j\leq p-1$. So far ...
3
votes
0answers
56 views

Determining whether or not an extension is a splitting field

On an exam for my modern algebra class, we were asked to determine whether or not $\Bbb{Q}\left(\alpha\right)$ is the splitting field (here using the convention that splitting field means the smallest ...
5
votes
1answer
86 views

A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields? My guess is yes, because every morphism ...
2
votes
1answer
39 views

Show the extension is not cyclic

Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
0
votes
2answers
84 views

Number of irreducible polynomials of degree $3$ over $\mathbb{F}_3$ and $\mathbb{F}_5$.

I'm trying to find the number of $3$rd degree irreducible polynomials over $\mathbb{F}_3$ and $\mathbb{F}_5$. Since a $3$rd degree polynomial is irreducible if and only if it is divisible by a ...
1
vote
1answer
100 views

Fixed field of automorphisms determined by $t\mapsto at+b$.

Suppose $E=\mathbb{F}_p(t)$, the field of rational functions in a transcendental $t$ over the finite field of $p$ elements. Suppose $G$ is the group of field automorphisms fixing $\mathbb{F}_p$ ...
1
vote
1answer
49 views

Irreducibility over Fp - A useless hint?

this is my first question here so bear with me. Dummit and Foote 13.5.5: For any prime p and nonzero $a \in \mathbb F_p$ prove that $x^p-x+a$ is irreducible and separable over $\mathbb F_p$ The ...
11
votes
3answers
143 views

Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?

When proving the Fundamental Theorem of Algebra, we need to appeal to analytic and/or topological properties of $\Bbb{C}$ and $\Bbb{C}[z]$. Is this going to be necessary in general, and if so, to what ...
4
votes
2answers
76 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
1
vote
2answers
68 views

Why can't a field have a subring which is isomorphic to $\mathbb{Z}/12\mathbb{Z}$?

Let $F$ be a field. Explain why $F$ cannot have a subring which is isomorphic to the ring $\mathbb{Z}/12\mathbb{Z}$. How to prove this?
1
vote
1answer
75 views

Fixed field of group of automorphisms on $\mathbb{C}(t)$.

I've been stuck on this problem tonight. Suppose $E=\mathbb{C}(t)$ where $t$ is transcendental over $\mathbb{C}$, and let $\omega$ be a primitive cube root of unity. Let $\sigma$ be the automorphism ...
6
votes
1answer
78 views

Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.

I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
3
votes
2answers
152 views

How to find the splitting field and Galois group of $x^6 -4x^3 +1$?

I am trying to find the splitting field $L$ of the $x^6 -4x^3 +1$ over $\mathbb{Q}$, and its Galois group. Here are some things I have figured out. I did the usual trick of solving for $x^3 = 2\pm ...

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