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Questions tagged [field-theory]

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. This tag is NOT APPROPRIATE for questions about the fields you encounter in multivariable calculus or physics. Use (vector-fields) for questions on that theme instead.

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Two different embeddings $ \mathbb R \to \mathbb C $ over $ \mathbb R $?

How to find all embeddings $ \mathbb R \to \mathbb C $ over $ \mathbb R $? I have found only trivial one $$ x \mapsto x + 0i, $$ but from theory it is known that $ [\mathbb C : \mathbb R]_s = \#\{\...
tensorix's user avatar
1 vote
0 answers
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Infinite Galois Correspondence on Ramakrishnan & Valenza

I'm reading the book Fourier Analysis on Number Fields by Ramakrishnan & Valenza (Theorem 1-20 on page 34). I'm a little confused by the discussion of infinite Galois theory. Let $K/F$ be Galois (...
Sardines's user avatar
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1 answer
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Galois group of $X^3-X+1$ over $\mathbb{Q}$ and $\mathbb{R}$ without discriminant.

Yesterday I had an exam and I had to find the galois group of the polynomial $f = x^3-x+1$. My answer was $A_3$ which is probably wrong. First of all it has no roots by the rational root theorem so it ...
muhammed gunes's user avatar
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1 answer
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Splitting Field of $x^4 - 10$ in $\mathbb{F}_7$

I am currently studying field theory and came across a problem that I need some help with. Specifically, I am interested in finding the splitting field of the polynomial $x^4 - 10$ over the finite ...
Khaled Alekasir's user avatar
1 vote
0 answers
24 views

A criterion for proving that an algebraic extension $E \subset F$ of fields is normal.

Let $K \subset F$ be fields such that $F$ is an algebraic extension of $K$, if for all elements $\alpha \in F$ there is a field $K \subset E \subset F$ such that $\alpha \in E$ and $E$ is normal over $...
Donlans Donlans's user avatar
2 votes
0 answers
65 views

$\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity

I need to prove/disprove the following statement: Every algebraically closed field $K$ contains a 10th root of unity. I don't think the statement is true. My counterexample is as follows: Let's take ...
muhammed gunes's user avatar
2 votes
0 answers
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A subfield $\mathbb{C}(u(t)) \subseteq \mathbb{C}(t)$ satisfying $\mathbb{C}(u(t),t^n)=\mathbb{C}(t)$

Let $u=u(t) \in \mathbb{C}[t]$, with $\deg_t(u)=d \geq 1$. Let $R=\mathbb{C}(u(t)) \subseteq \mathbb{C}(t)$ be the subfield generated by $u$. It is clear that $[\mathbb{C}(t):R]=d$. For every $n \geq ...
user237522's user avatar
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1 answer
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Linear space and vector space correlation

I'm confused with definition of vector space and field. According to wiki vector_space and field. Vector space over some field is defined as set of element in $V$ and binary operations that satisfies: ...
Pyrettt Pyrettt's user avatar
4 votes
2 answers
109 views

Alternative method : $\mathbb{Q}(\zeta_3+\sqrt[3]{7})=\mathbb{Q}(\zeta_3,\sqrt[3]{7})$

Let $\zeta=\zeta_3$ be a third root of unity. I want to proof that $\mathbb{Q}(\zeta+\sqrt[3]{7})=\mathbb{Q}(\zeta,\sqrt[3]{7})$. One inclusion is clear that is : $\mathbb{Q}(\zeta+\sqrt[3]{7})\subset\...
muhammed gunes's user avatar
-1 votes
1 answer
47 views

For certain field extensions $[L_1:F]<\infty$, $[L_2:F]<\infty$, $L_1 \simeq L_2$, when $[L_1:F]=[L_2:F]$?

Let $\mathbb{C} \subset F \subseteq L_1 \subseteq \bar{F}$, $\mathbb{C} \subset F \subseteq L_2 \subseteq \bar{F}$ be field extensions, with $[L_1:F]=n_1<\infty$ and $[L_2:F]=n_2<\infty$, $\bar{...
user237522's user avatar
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-1 votes
1 answer
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The $n$-th root in the ring of polynomial over a field containing $\mathbb{C}$

Let $K$ be a field containing the complex number field $\mathbb{C}$, and $f\in K[X_{1},\dots,X_{n}]$ a polynomial with $n$ variables. Suppose $L$ is an extension field of $K$ and $f=g^{m}$ for some $m\...
Waker's user avatar
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0 answers
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What is the Galois Group of $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?

I am studying for some algebra qualifying exams over the summer and I am stumped on the title question: What's the Galois Group for $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$? Here's what I've got so far: ...
S.H.'s user avatar
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1 answer
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Why cyclic subgroup of $\text{Gal}(L/K)$ is achieved by some decomposition group?

Let $L/K$ be a finite Galois extension. Let $\text{Gal}(L/K)$ be its Galois group. For every cyclic subgroup $H$ of $\text{Gal}(L/K)$, why does there exists a prime $v$ of ring of integers of $K$ and ...
Poitou-Tate's user avatar
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Concerning some quadratic field extension $[L:\mathbb{C}(u,v)]=2$

In the following question I am actually asking about the answer to this MO question. First, I will ask it generally, then I will present the question and answer appearing in MO. A general question: ...
user237522's user avatar
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1 vote
0 answers
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Conway Polynomial for p=2, n=3?

Im doing an exercise on Conway polynomials. As far as im concerned, for p=2, n=3 both $f(x)=x^3 + x^2 + 1$ and $g(x)=x^3 + x + 1$ satisfy every condition. According to every source i found, the latter ...
Vanessa K's user avatar
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0 answers
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How to show that the field $\mathbb{F}_2(t^{\frac{1}{3}})$ does not contain a third root of unity

I want to find a non-normal extension in characteristic p. The following extension is not normal : $$\mathbb{F}_2(t)\subset \mathbb{F}_2(t^{\frac{1}{3}})$$ It's minimal polynomial is : $X^3-t$ and it'...
muhammed gunes's user avatar
0 votes
1 answer
111 views

Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
user21820's user avatar
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3 votes
2 answers
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Let $E=\mathbb{Z}_3[x]/\langle x^2+x+2\rangle$. How many elements of order 2 and 3 are there in the additive group of $E$? How many generators?

I know that for $E=\mathbb{Z}_3[x]/\langle x^2+x+2\rangle $, since $x^2+x+2$ is irreducible over $\mathbb{Z}_3$, $E$ is a field, because $\langle x^2+x+2\rangle $ is a maximal ideal. In addition, $E$ ...
Camilo Diaz's user avatar
2 votes
1 answer
71 views

An exercise about abelian Kummer extensions

I'm trying to do this problem about abelian Kummer extensions: Image transcript and my attempts are below: Let $K/F$ be a Galois extension with Galois group $G=\operatorname{Gal}(K/F)$ of order $n$. ...
hbghlyj's user avatar
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1 vote
0 answers
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Understanding a proof in JS Milne's Fields and Galois Theory (Prop 7.10)

The following is proposition 7.10 in Milne's Fields and Galois Theory: Let $G$ be a group of automorphisms of a field $E$, and let $F=E^G$ (ie. $F$ is fixed field of $G$). If $G$ is compact and the ...
Ajin Shaji Jose's user avatar
2 votes
0 answers
57 views

Proving $ 16a^2+20\sqrt{2}a+15=\sqrt{11(a^{12}-1)}+\sqrt{2(a^{12}-4)}$ for a specific algebraic number $a$

Let $F=\mathbb{Q} (\sqrt{2})$ and let $a$ be the real root of $f(x) =x^3-2x-\sqrt{2}\in F[x] $. Using Cardano's approach one can show that $$a=\sqrt[3]{\frac{1}{\sqrt{2}}+\sqrt{\frac{11} {54}}}+\...
Paramanand Singh's user avatar
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2 votes
1 answer
61 views

If $a \in \mathbb{C}$ is algebraic over $\mathbb{Q}$, determine the minimal polyn of $\sqrt{a}$ over $\mathbb{Q}$ when $\sqrt{a} \notin \mathbb{Q}(a)$

This exercise is broken down into parts: I first proved that $\sqrt{a}$ is algebraic over $\mathbb{Q}$. Given that $a \in \mathbb{C}$ is algebraic over $\mathbb{Q}$, let $m_a(x)$ be the minimal ...
Camilo Diaz's user avatar
-3 votes
0 answers
26 views

Irreducible polynomial over a finite field is irreducible in $\mathbb{Z}$. [closed]

Let $f\in\mathbb{Z}[x]$ be a monic polynomial of degree 5. Furthermore suppose that $p$ is a prime number and that $F$ is a finite field of order $p^2$ such that $f$ has no roots (in $F$). Show that $...
Thora N's user avatar
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0 answers
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$k(x)$ is finitely generated as a $k$-algebra, can I view $k(x)/k$ as a field extension?

I try to prove that the finite dimensional division algebra $\Delta$ over algerbaically closed field $k$ is $k$ itself. First, we take $0\neq x\in\Delta$ then consider $k(x)\subset\Delta$ as ...
wwwwww's user avatar
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1 answer
71 views

How to prove that $[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}]=12$

I know that $$[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}]=[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}(3^{1/4})]\cdot[\mathbb{Q}(3^{1/4}):\mathbb{Q}],$$ and $$[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}]=[\mathbb{...
Camilo Diaz's user avatar
1 vote
0 answers
27 views

An algebra structure over a perfect field

When I read some paper, I see the following word: When $k$ is a perfecr field, a finite dimension $k$-algebra $A$ can be writted as $S\oplus \rm{rad}(A)$, where $S$ is the maximal semisimple ...
Zhenxian Chen's user avatar
8 votes
1 answer
264 views

Isomorphism of topological groups

Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
ByteBlitzer's user avatar
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0 answers
35 views

Primitive elements of the rationals function extension [duplicate]

I want to prove this: Let $k$ a field, $E:=k(x)$ the quotient field of $k[x]$ and $u\in E$. Show tha $E=k(u)$ if, and only if, $u=\frac{ax+d}{cx+d}$ for some $a,b,c,d\in k$ such that $ad-bc\neq 0$. I ...
Diego Ramírez's user avatar
0 votes
1 answer
79 views

"field" vs. "vector field" [duplicate]

Is the "field" in the "vector field" as the same "field" in algebra: as the commutative ring with the multiplicative inverse? If yes, then the "vector field" ...
wonderich's user avatar
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2 votes
1 answer
111 views

Irreducibility of a Polynomial with Prime Exponents

Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$ where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
math.enthusiast9's user avatar
1 vote
0 answers
33 views

field lines of a $\mathbb{R}^2 \to \mathbb{R}^2$ function (ellipses) [closed]

i have the following exercise: Characterize the field lines of the following vector field $g: \mathbb{R}^2 \to \mathbb{R}^2:$ $g(x,y)=(\frac{-y}{a^2}, \frac{x}{b^2}) $ with $a,b \in \mathbb{R} \...
D P's user avatar
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0 answers
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Proof of the existence of algebraic closure

I am taking an introductory course on Galois theory, and we recently covered the following theorem: For every field $\mathbb{F}$, there exists a field extension $\mathbb{L}$ which is algebraically ...
A.P's user avatar
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1 vote
0 answers
35 views

Let $E$ be an extension of $F$ and let $A = \{e \in E: e$ is algebraic over $F\}$. Show that $A$ is a subfield of $E$ containing $F$.

Let $E$ be an extension of $F$ and let $A = \{e \in E : e$ is algebraic over $F \}$. Show that $A$ is a subfield of $E$ containing $F$. First, clearly $A \subseteq E$. Also since $F$ is algebraic over ...
Grigor Hakobyan's user avatar
3 votes
1 answer
93 views

Prove that $[\mathbb{Q}(\sqrt{3}, \sqrt[3]{3}, \sqrt[5]{3}, \xi_3,\xi_5) \colon \mathbb{Q}] = 240$.

I have already proven that $[\mathbb{Q}(\sqrt{3}, \sqrt[3]{3}, \sqrt[5]{3}) \colon \mathbb{Q}] = 30$ and $\sqrt{3} \not\in \mathbb{Q}(\xi_5)$. Consequently, $[\mathbb{Q}(\xi_5,\sqrt{3}) \colon \...
David's user avatar
  • 165
0 votes
0 answers
32 views

Does $Z(f) \cap Z(g) = \emptyset$ implies $\gcd(f,g) = 1$? [duplicate]

If $f(x)$ and $g(x)$ are in $\mathbb{F}[x]$, where $\mathbb{F}$ is a field. Let us assume that $Z(f) = \{x \in \overline {\mathbb{F}} : f(x) = 0\}$, where $\overline{\mathbb{F}}$ is the algebraic ...
Afntu's user avatar
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0 votes
3 answers
77 views

How to prove $\sqrt{2} \notin \mathbb{Q}(\sqrt[3]{2})$ using elementary techniques?

Prove $\sqrt{2} \notin \mathbb{Q}(\sqrt[3]{2})$. My try by contradiction: Assume there exist $a,b,c \in \mathbb{Q} $ such that: $\sqrt{2}=a + b \sqrt[3]{2} +c \sqrt[3]{4}$. Squaring both sides, we ...
Mahmoud albahar's user avatar
2 votes
2 answers
86 views

$\mathbb{F}_{11}[x]/(x^2+1)$ and $\mathbb{F}_{11}[y]/(y^2+2y+2)$ are both fields with 121 elements

Below is my attempt to prove that $\mathbb{F}_{11}[x]/(x^2+1)$ and $\mathbb{F}_{11}[y]/(y^2+2y+2)$ are both of order $121$: Notice that $\mathbb{F}_{11}[x]/(x^2+1)$ are a field with $\{1, x, x^2, x^3,\...
End points's user avatar
1 vote
1 answer
131 views

If $\alpha \in \mathbb R$ is algebraic over $\mathbb Q$, then there exists $n \in \mathbb N$ such that $\alpha^n \in \mathbb Q$. [duplicate]

I wish to disprove the following proposition: If $\alpha \in \mathbb R$ is algebraic over $\mathbb Q$, then there exists $n \in \mathbb N$ such that $\alpha^n \in \mathbb Q$. I'm trying to give a ...
RatherAmusing's user avatar
2 votes
1 answer
65 views

Subfields of splitting field of $x^4+25$ over $ℚ$.

Let $F$ be the splitting field of the polynomial $x^4+25$ over $ℚ$. List all subfields in $F$ and the corresponding subgroups in the Galois group. is problem $1$ on this pdf. The solution is: As we ...
hbghlyj's user avatar
  • 3,035
0 votes
0 answers
38 views

If $\alpha$ is algebraic over $F$, then it is algebraic over any extension field $L$ of $F$.

The photo attached is from the book Abstract Algebra by Dummit&Foote. I didn’t understand it at all. How do we know that $L$ contains the element $\alpha$. In other words, what does $\alpha $ have ...
boyler's user avatar
  • 375
0 votes
1 answer
49 views

Proof that the finite-dimensional extension $K[a]$ coincides with the decomposition field $K(a)$

Let $K\subseteq F$ be an extended field and $a\in F$. Let $K[a]=\left \{ f(a)|f\in K[x] \right \}$. Prove that if $K[a]$ is finite-dimensional as a vector space over $K$, then $K[a]=K(a)$ Consider ...
Dmitry's user avatar
  • 1,362
2 votes
1 answer
89 views

If $F(\alpha) = F(\alpha^2)$, then $\alpha$ is algebraic over $F$

Let $K$ be an extension of a field $F$ and let $\alpha \in K$. If $F(\alpha) = F(\alpha^2)$, then $\alpha$ is algebraic over $F$. I have an idea about how to do this that doesn't trivialize the ...
Grigor Hakobyan's user avatar
0 votes
0 answers
25 views

Proof of equivalent definitions of normal extensions

Let $F \supset k$ be an algebraic extension. We have these two characterizations of a normal extension: a) For every $k$-homomorphism $\sigma: F \to \overline{k}$, we have $\sigma(F) \subset F$ b) If ...
lkksn's user avatar
  • 91
0 votes
1 answer
25 views

Splitting fields and isomorphisms

If $K \subseteq L$, $K \subseteq L'$ are field extensions, $L \cong L'$ and $L$ is the splitting field of $f \in K[x]$, is $L'$ also a splitting field of $f$ ? I think $L'$ is a splitting field $\iff$ ...
ed268's user avatar
  • 71
0 votes
0 answers
65 views

Reducibility of $x^2-7$ over $\mathbb{Q}(\sqrt[5]{3})$

Suppose for a contradiction that $x^2-7$ is reducible over $\mathbb{Q}(\sqrt[5]{3})$. Then $\sqrt{7}\in\mathbb{Q}(\sqrt[5]{3})$. It follows that $\mathbb{Q}\subset\mathbb{Q}(\sqrt{7})\subset\mathbb{Q}(...
spinosarus123's user avatar
1 vote
0 answers
57 views

Question in an example of prime decomposition in an extension of DVRs.

Choose any element $\alpha \in \mathbb{Q}[[x]]$ such that it is transcendental over $\mathbb{Q}(x)$(This is possible by comparing cardinalities.), and let $\beta=\alpha^2$. Let $K=\mathbb{Q}(x,\beta)$,...
clgdj's user avatar
  • 159
0 votes
0 answers
48 views

How to prove that two algebraically closed field with same characteristic and cardinal are isomorphic? [duplicate]

Hi I would like to prove that for algebraic closed field $F_1,F_2$ if: They have the same characteristic $p$ They have the same cardinal and is uncountable Then they are isomorphic. I have proved ...
Shore's user avatar
  • 343
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0 answers
48 views

How to prove expended fields whose base has same cardinal are isomorphic?

I would like to prove that for field $F_1,F_2$ if they : have isomorphic sub-field $F'$ $F_1,F_2$ is the vector space over $F'$, and their bases, doneta $B_1,B_2$, has the same cardinal Then $F_1,...
Shore's user avatar
  • 343
2 votes
1 answer
68 views

Find a counterexample: for $a,b \in K \setminus\{0\}$, equation $ax^2 + b y^2 = 1$ has a solution iff $ax^2 + by^2 = z^2$ has a nontrivial solution

It is a well-known fact that if $\text{char}(K) \neq 2$ and $a,b \in K \setminus\{0\}$, then the equation $ax^2 + by^2 = 1$ has a solution over $K$ if and only if the homogeneous equation $ax^2 + by^2 ...
Neckverse Herdman's user avatar
1 vote
0 answers
29 views

Order of $\mathbb F _p [x] / (f)$.

I could use some help with the following exercise: Find the number of reducible monic polynomials of degree $2$ over $\mathbb F_p$. Show this implies that for every prime $p$ there exists a field of ...
RatherAmusing's user avatar

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