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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0
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2answers
36 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 ...
1
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2answers
14 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 ...
15
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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 ...
1
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2answers
28 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 irreduciable in $\mathbb{Q}[X]$, ...
0
votes
1answer
20 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 ...
1
vote
0answers
19 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 ...
1
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2answers
69 views

If $E,F$ are finite fields and $F\subseteq E,$ why is $E$ a finite-dimensional vector space over $F$?

I understand that if $E$ and $F$ are each finite and $E$ is a vector space over $F$, then $E$ must be a finite-dimensional vector space over $F$. However, my question is: why does $F\subseteq E$ imply ...
0
votes
2answers
32 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 ...
0
votes
0answers
27 views

Minimal polynomial (Field theory)

Let $\alpha = \sqrt[4]{2}$. My question is: ${\rm Polmin}(\alpha,\mathbb{Q}(i))=X^4-2$? Because: $${\rm Polmin}(\alpha,\mathbb{Q}(i)) \mid {\rm Polmin}(\alpha,\mathbb{Q})=X^4-2.$$ But ...
0
votes
1answer
19 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?
1
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1answer
11 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
66 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
19 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 $o(\bar x^2)=2$ ...
1
vote
1answer
22 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
vote
1answer
20 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 ...
0
votes
1answer
13 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 ...
1
vote
2answers
63 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 ...
1
vote
2answers
52 views

Irreducible polynomial over Q

Let $f(x) = 3x^4+6x^3+24x^2+18 \in \mathbb Z[x]$. Is $f(x)$ irreducible over $\mathbb Q$ ? In my course, Eisenstein's criterion is apply for monic polynomial only, hence, I can't use it with p =2. If ...
0
votes
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 ...
0
votes
2answers
20 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 ...
3
votes
2answers
48 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
votes
2answers
24 views

Square root constructible elements in a splitting field. [closed]

Let α have minimal polynomial p(x) ∈ $\mathbb{Q}$[x], with roots α = α1, ..., αs. Let K = $\mathbb{Q}$(α1, ..., αs) be the splitting field of p(x). Let [K:Q] = 2w for some integer w. Prove ...
2
votes
2answers
24 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 ...
0
votes
1answer
21 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
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 ...
3
votes
0answers
40 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 ...
8
votes
2answers
2k views

Degree of $\sqrt{2}+\sqrt[3]{5}$ over $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{5})$

I'm self-studying field extensions. I ran over an exercise which I can't completely solve. (I haven't yet started studying Galois theory, and I think this exercise isn't meant to be solved using it, ...
2
votes
2answers
88 views

Suppose that $a$ and $b$ belong to a field of order $8$ and that $a^2 + ab + b^2 =0$ then $a=0$ and $b=0$ . [duplicate]

Suppose that $a$ and $b$ belong to a field of order $8$ and $a^2 + ab + b^2 =0$. Then $a=0$ and $b=0$. Do the same when the field has order $2^n$ with $n$ odd? If one of the term is zero, i.e. ...
4
votes
6answers
127 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 ...
1
vote
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 $x=2^{\frac{1}{2}}+5^{\frac{1}{3}}$, $x-2^{\frac{1}{2}}=5^{\frac{1}{3}}$, ...
0
votes
0answers
43 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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votes
0answers
9 views

Existence of non-dedekind-complete archimedean field?

I'm a first year undergrad who just started learning analysis in the 2nd semester, so please do forgive me if I made any terrible mistake in my question. Anyway, I was reading some introductory ...
0
votes
1answer
37 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 ...
2
votes
1answer
24 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 ...
3
votes
2answers
52 views

Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field ...
0
votes
1answer
32 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 ...
1
vote
0answers
29 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 $). ...
1
vote
1answer
53 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, ...
0
votes
1answer
40 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
votes
2answers
66 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 ...
1
vote
1answer
56 views

Degree of field extension $F(x) / F(x^2 + 1 / x^2)$

Let $y=\frac{x^4+1}{x^2} \in F(x)$. Then $g(x)=0$ for the polynomial $g(s) = (s^4+1)-ys^2$. How to show that it is the minimal polynomial over the field $F(y)$?
2
votes
2answers
36 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 $K \subseteq Z(D)$?
5
votes
3answers
59 views

Degree of an extension of $ \mathbb{Q} $

I'm trying to find a degree of the extension $ \mathbb{Q} \subset \mathbb{Q}(\sqrt{2} + i) $. Once I'm done with that, I'd like to find a basis of $ \mathbb{Q} (\sqrt{2} + i) $ as a $ \mathbb{Q} $ ...
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votes
0answers
36 views

Algebraic extensions help?

$K$ is an extension field of $F$. If $[K : F]$ is finite and $u$ is algebraic over $K$, prove that $[F(u) : F]$ divides $[K(u) : F]$.
4
votes
3answers
93 views

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ a number field?

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ of $5$-adic numbers a number field, if yes what is the degree ? To be honest I don't understand the question, what does it mean ...
0
votes
1answer
23 views

Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.

Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series. $F[[x]]$ is contained in $F((x))$. So there's at least a ring homomorphism that is injective. Can also see ...
1
vote
2answers
59 views

$i \notin \mathbb{Q}[\sqrt[4]{2}]$ without using topological properties of $\mathbb{R}$

I can think of two related ways to prove that $i \notin K = Q[\sqrt[4]{2}]$: $K$ is a subset of the real numbers and $i$ is not a real number. $K$ is orderable and no ordered field can contain ...
1
vote
1answer
65 views
3
votes
1answer
80 views

Let $K$ a field with characteristic $p>0$. Show that $\{x \in K : x^{p^n} =x \}$ is a subfield.

Let $K$ a field with characteristic $p>0$. I've shown that for every positive $n$ the set $\{ x^{p^n} : x \in K \}$ is a subfield of $K$, I did this by showing that $F:K\to K: x \mapsto x^{p^n}$ is ...
7
votes
2answers
163 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...