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. Please use (galois-theory) instead for questions specifically about that ...

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32 views

Degree $5$ Irreducible Polynomial Solvable by Radicals and Abelian Extension

Consider an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is an ...
2
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1answer
41 views

Proving that the unit cube cannot be tripled (with straight edge and compass)

I would like to show the unit cube cannot be tripled using a straightedge and compass. I note that the side of a cube that has been tripled would have a side length of $\alpha=\sqrt[3]{3}$ But ...
2
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1answer
61 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, but $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not

How can I show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ but that $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not? I am kind of lost with Galois Theory. Thanks
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0answers
32 views

Identifying a group

When asked to identify the group $\mathbb{F}_4^{+}$, is my explanation below complete? If not, how can I complete it? The field $\mathbb{F}_4^{+} = \mathbb{F}_2[x]/(x^2+x+1)$ consists of the residues ...
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3answers
34 views

Finite fields and isomorphism

For each prime number p, let $F_p$ denote the field of integers modulo p. Now let K be any finite field. a) Prove that K contains a subfield isomorphic to $F_p$ for some prime number p b) Prove that ...
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2answers
56 views

Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

I do not understand why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$. I know that $x^{4}+1$ is irreducible $(f(x + 1) = (x + 1)^4 + 1$ is Eisenstein at $2$) and it has the roots: ...
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1answer
32 views

Galois group of $f$ is cyclic if $\deg f$ is prime

Hello I am learning Galois Theory by myself and got lost in the following exercise: Let $f$ be an irreducible polynomial of degree $n$, and suppose that the splitting field of $f$ is generated by a ...
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0answers
29 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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1answer
37 views

Squares in a field with $q^n$ elements.

Let $F$ be a field with $q^n$ elements, where $q$ is an odd prime. Write $q^n=2m +1$ with $m \in \mathbb{N}.$ If $r \in F^{\times},$ show that the equation $y^2= r$ has a solution iff $r^m=1.$
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2answers
31 views

How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
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0answers
62 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
2
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1answer
32 views

$a$ and $1+a^{-1}$ have same degree over $F$ if $a$ is algebraic over $F$

Suppose that $a$ is algebraic over a field $F$. Show that $a$ and $1+a^{-1}$ have the same degree over $F$. Not really sure where to start for this one. I know that I have to show that ...
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1answer
12 views

What is an example of $E/F,L/E$ are normal but $L/F$ is not. [duplicate]

Let $E/F,L/E$ be normal field extensions. What would be an example such that $L/F$ is not normal?
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2answers
37 views

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$.

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$. Let $f(x)=x^2+x+1$. Then $f(a)=a^2+a+1=0$. To show equality of the two fields, we need ...
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2answers
65 views

How to find the degree of an extension field?

How to find the degree of an extension field ? Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ? I've ...
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2answers
15 views

What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
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2answers
38 views

How do I find all conjugates of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$?

How do I find all conjugates of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$? I have shown that $\sqrt{1+\sqrt{2}}$ is a root of $X^4 - 2X^2 - 1$ and this polynomial is irreducible in $\mathbb{Q}[X]$, ...
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2answers
47 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
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1answer
68 views

Showing that $40^{\circ}$ is not constructible

Show that $40^{\circ}$ is not constructible. Attempt We note that $\cos 120^{\circ}=-\frac{1}{2}$ and that it also equals $4\cos^340^{\circ}-3\cos40^{\circ}$, which is obtained by using the ...
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0answers
20 views

What would be a value of $X$ under an automorphism of $F(X)$ over $F$?

Let $\sigma:F(X)\rightarrow F(X)$ be a field automorphism fixing $F$. What would be an value of $\sigma(X)$? Since $X$ is transcendental over $F$, $\sigma(X)$ is a transcendental over $F$ and ...
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1answer
29 views

Q basis for splitting field

I have the following field theory question: I am given this polynomial $ x^5-5 $ for which I am supposed to find a basis for the splitting field over Q all I can determine in this regard is that it ...
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1answer
20 views

prove $[E(a):E] \le [F(a):F]$

Let $F<E<K$ be field extensions, such that $a \in K$, and $[K:F]<\infty$, Is it true that $[E(a):E] \le [F(a):F]$? How can I show this?
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1answer
17 views

Are these two definitions of separable extension equivalent?

Definition1 - wikipedia Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable. Definition 2 - ...
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2answers
77 views

Is there an infinite field such that every non-zero element has finite multiplicative order?

Is there an infinite field such that every non-zero element has finite multiplicative order? I did not find any example of such a field, but also did not see anything that forbids the existence of ...
2
votes
1answer
25 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
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1answer
28 views

Question about field extension notation

Hello all I was given the following question about which I understand everything except possibly the notation. I am given a sub-field $ F \subseteq R $ and I am asked to prove the degree of the "field ...
1
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1answer
21 views

Prove that there are no separable extensions of $k$ of degree $n$

Let $k$ be a field and let $n \gt 0$ be an integer. Assume that there are no irreducible polynomials of degree $n$ in $k[x]$ . Prove that there are no separable extensions of $k$ of degree $n$ I ...
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2answers
2k views

Proving the “freshman's dream”

$(x+y)^p = x^p + y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems. What is the ...
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2answers
22 views

Show that if $F$ is a field, then $<x>$ is maximal in $F[x]$. Also, show that $F[x]$ is not local.

See statement above. So far I have the following: Assume that $<x>$ is not maximal. Then $ <x> \subset <f(x)> \neq F[x]$. This means that $x = f(x) g(x)$. Since $x$ is ...
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1answer
14 views

About relation between degree of extension and normality

Hi i know that if $F<E$ is an field extension and $[E:F]=2 $ then it is normal extension. How can i show that if $[E:F]=k$ for any $k>2 $ doesnt imply E is normal extension. Basically i need ...
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2answers
66 views

Field $K (x)$ of rational functions with coefficients from $K$, if $f\in K(x)$, then $f^2 \neq x^2-1$

I'm in the process of studying for an exam and I came across the following question: Prove that if $K$ is a field and $K (x)$ is the field of rational functions with coefficients from $K$, if ...
3
votes
2answers
50 views

Prove that $x^2 + 3x +2$ is irreducible in $\mathbb{Z}[[x]]$, but not in $\mathbb{Z}[x]$.

As the problem states, I need to show irreducibility of the given polynomial. I'm not sure where go with this, so any help would be great. I know that Eisenstein has a nice test for this in ...
0
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1answer
27 views

How's this inertia called?

Let $E/F$ be an algebraic extension. Let $L_1,L_2$ be algebraically closed fields and $\sigma_1:F\rightarrow L_1,\sigma_2:F\rightarrow L_2$ be field monomorphisms. Define ...
2
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2answers
25 views

Does every algebraically closed field with nonzero characteristic have a unique finite subfield $p^n$?

Let $F$ be an algebraically closed field such that $char(F)\neq 0$. Then, $\forall n\in\mathbb{Z}^+$, there exists a unique finite subfield $K$of $F$ such that $|K|=char(F)^n$. Is this ...
3
votes
0answers
53 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
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0answers
45 views

Questions about the function fields of complex algebraic surfaces

Let $X$, $Y$ be complex algebraic surfaces(Of course, they are smooth). Suppose that $X$ is normal. Let $K(X)$ and $K(Y)$ be the function fields of $X$ and $Y$, respectively. And we have a ...
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0answers
43 views

A problem about an algebraic number [duplicate]

Show that $2^{\frac{1}{2}}+5^{\frac{1}{3}}$ is algebraic over $\mathbb{Q}$ of degree $6$. Can I just construct $x=2^{\frac{1}{2}}+5^{\frac{1}{3}}$, $x-2^{\frac{1}{2}}=5^{\frac{1}{3}}$, ...
2
votes
1answer
19 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
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6answers
132 views

Is $\mathbb{Q}(\sqrt{2},\sqrt[3]{5})=\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$?

I understand that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ But I am struggling to algebraically show that ...
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2answers
39 views

Definition of a Finite Normal Extension

I'm reading Fraleigh's A First Course in Abstract Algebra, Seventh Edition, and he makes the following definition on page 448: A finite extension $K$ of $F$ is a finite normal extension of $F$ if ...
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1answer
45 views

Finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field $\mathbb{Q}(\zeta_3).$

How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of ...
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1answer
26 views

Jets and vertical differential

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
2
votes
1answer
26 views

Finite field extensions - $K(\alpha)$

So I am currently studying Algebraic Number theory and a theorem in the Book states the following: Let $L/K$ be a field extension. Then $\alpha \in L$ is algebraic over $K$ if and only if there is ...
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1answer
38 views

Finite field extension over the rationals need only one generator?

Some book stated (without proof) that in every finite dimensional field over the rationals of dimension $n$, there is an element of degree $n$ (i.e. any field $Q[\alpha_1, ..., \alpha_n]$ is of the ...
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1answer
43 views

Degree of minimal polynomial over $\mathbb{Z}_7$

While working through my book I've run into a question where I'm not too sure what is being asked of me/how to start thinking about it. It states: Suppose $E$ is an extension field of ...
2
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2answers
70 views

Prove that $\mathbb{Q}(r+s\sqrt{t})=\mathbb{Q}(\sqrt{t})$.

Let $r,s,t\in\mathbb{Q}$. Prove that $\mathbb{Q}(r+s\sqrt{t})=\mathbb{Q}(\sqrt{t})$. Ok. So I've fallen a little behind in my algebra class, and I'm a bit confused on how to approach this ...
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1answer
62 views

Annihilating Ideal of a Ring

I am stuck on how to show this. A starting hint would be helpful, and an answer (hidden) would be much appreciated. I tried supposing that there was another element in the annihilating ideal, however, ...
3
votes
4answers
131 views

Book recommendation for Abstract Algebra

Hello fellow members of Math StackExchange! Since this is my first post here, I should start with a little introduction of myself. I've recently finished the second year of math at university and I ...
2
votes
2answers
39 views

algebraically closed field in a division ring?

Is it possible to have $K \subset D$ where $K$ is algebraically closed field and $D$ is a division ring such that $K \subseteq Z(D)$?
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0answers
30 views

Field extension in rational functions [duplicate]

I'm facing the following problem: Let $ F $ be a field, and let $ F(x) $ denote all rational functions over $ F $ (functions of form $\frac{P(x)}{Q(x)}$, where $ P,Q$ are polynomials over $ F $). ...