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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1
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1answer
46 views

If $x+y\sqrt{n} \in \mathbb{C}$ is a root of $f$ then $x-y\sqrt{n}$ is also a root

Let $n\in \mathbb{Z}$ be a non-square integer and $x+y\sqrt{n} \in \mathbb{C}$ a root of $f\in \mathbb{Q}[x]$ with $x,y\in \mathbb{Q}$. Show that $x-y\sqrt{n}$ is also a root of $f$. To show ...
0
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1answer
26 views

Algebraically closed field and polynomials

There is the problem: Let $F$ be a field of characteristic $0$ with the condition: If $f(x) \in F[x]$ has no roots in $F$, then the degree of $f(x)$ is a multiple of $21$. Prove that $F$ is ...
1
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2answers
33 views

How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
2
votes
1answer
49 views

Additive combinatorics and $|S \cap (S+t)| \geq K |S|$ over $\mathbb{F}_2^n$

I don't know much about additive combinatorics, and I am wondering if there are results concerning the size of $|S \cap (S+t)|$ where $S \subset \mathbb{F}_2^n$ and $t \in \mathbb{F}_2^n$. Especially, ...
1
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0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
5
votes
0answers
79 views

Show that a sum of consecutive radicals is irrational $\forall n$ [closed]

I need to show that the number $\sqrt 2+ \sqrt[3]{3}+\sqrt[4]{4}+\sqrt[5]{5}+\cdots+\sqrt[n]{n}$ is irrational for any $n\ge2$. I don't have a clue about how I could show that. Thank you!
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1answer
37 views

Find the degree of the splitting field of a polynomial over $Q$ [closed]

I want to know how to determine the degree of the extension $K/\mathbb{Q}$, where $K$ is the splitting field of the polynomial $x^6+1\in \mathbb{Q}[x]$ over $\mathbb{Q}$. Do I have to get all the ...
1
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0answers
41 views

Find the value of $n$

Let $F$ be a field having $5^n$ elements .Also $F$ has an element which satisfies $x^{5^n}=1$ such that $x\neq 1$. Find $n$ . My try: Let $x\in F $ satisfy $x^{5^n}=1$ .Obviously the group ...
0
votes
1answer
35 views

Find the lattice of Galois Field

I am wondering what the lattice of subfield of $GF(p^{30})$ looks like. I know that it starts from $GF(p)$ and then $GF(p^2)$ and $GF(p^3)$, but then I am lost. And I looked it up online, but can't ...
2
votes
1answer
19 views

non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
1
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1answer
70 views

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Here's what I've thought so far. Unless I'm very much mistaken, we have that $[\mathbb{Q}(\sqrt{3},\sqrt{5}):\mathbb{Q}] = 15$, so I'm looking for a $\theta$ such that $[\mathbb{Q}(\theta):\mathbb{Q}] ...
0
votes
1answer
27 views

Show that any field K has a subfield isomorphic to either $\mathbb{Q}$ of $\mathbb{Z}_p$

Show that any field K has a subfield isomorphic to either $\mathbb{Q}$ of $\mathbb{Z}_p$ I understand that here we are talking about a prime subfield that would be isomorphic to either one or the ...
1
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2answers
28 views

Does every non-Archimedean ordered field contain $\mathbb Q (x)$ as an ordered subfield?

Every ordered field $\mathbb F$ contains $\mathbb Q$ in a canonical way. If the field is not Archimedean there exists an $x>n$ for all $n \in \mathbb N$. Since we are dealing with a field any ...
2
votes
2answers
38 views

Galois group of a quartic which is also a quadratic in $x^2$

A few weeks ago a professor of mine mentioned that the Galois group of a certain type of quartic polynomial is easy to calculate, and at the time it seemed obvious to me so I didn't ask why. Now i'm ...
2
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0answers
30 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
2
votes
2answers
31 views

Degree of splitting field of $x^6-2$ over $\mathbb{Q}$

Let $f=x^6-2$, find the degree of splitting field of $f$ over $\mathbb{Q}$. I calculated the roots of $f$ are $\pm \sqrt[6]{2},\pm e^{i \pi/3}\sqrt[6]{2},\pm e^{2i \pi/3}\sqrt[6]{2}$. I suspect ...
1
vote
1answer
37 views

how would you show that field automorphisms fix prime subfields?

Suppose K is a prime subfield of E, then if $\phi$ is an automorphism from E to E, we have for all x $\in$ K, $phi(x) = x$. I feel like this is just the definition of a field automorphism, but my ...
0
votes
2answers
26 views

Showing that the powers of a root of the p-th cyclotomic polynomial are distinct roots thereof.

Suppose $p$ is a prime number and $p(x) = x^{p-1} + x^{p-2} + ... + x + 1$ is the p-th cyclotomic polynomial and $\gamma$ is a root thereof, so that $p(\gamma) = 0$. I need to show that the following ...
1
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1answer
34 views

Degree of the extension field

Aren't we supposed to know the degree of the field extension to solve this problem? Did I miss something?
0
votes
1answer
37 views

Show that a polynomial is still irreducible in a extension field

I have found this question on the Papantonopoulou's Algebra book: Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ with deg$ f(x) = 15$ and deg$ g(x) = 14$. Let $\alpha$ be a ...
0
votes
1answer
45 views

Give an example of a polynomial whose Galois group is isomorphic to $D_{10}$

I have been spending my leisure time determining the subfield lattices and corresponding Galois subgroup lattices of some splitting fields of polynomials. I have made the lattice diagrams for the ...
2
votes
0answers
71 views

Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
11
votes
1answer
120 views

Surjective exponentials for algebraically closed fields

The existence of the exponential on $\mathbb{C}$ has a very basic, yet very strong consequence : $(\mathbb{C}^*,\cdot)$ is a quotient of $(\mathbb{C},+)$. This question is concerned with fields $K$ ...
0
votes
2answers
27 views

For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Let us define numbers $a_i\in\mathbb{R}_{\geq0}$ by $a_0=0$ and $a_{i+1}=\sqrt{2+a_i}$. How do I prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$? How do I prove that ...
2
votes
3answers
74 views

Showing $\mathbb{Q} \times \mathbb{Q}$ is not a field

I am revising and have come across the question Show that $\mathbb{Q} \times \mathbb{Q}$ with element-wise addition and multiplication is not a field I don't understand how to go about this, do i ...
2
votes
2answers
69 views

Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
3
votes
1answer
35 views

On the degree of cyclotomic fields extension.

Let $F$ be a field and $n$ be any integer satisfying $(n,\text{char}(F))=1$ when $\text{char}(F)\ne 0$. Let $\xi$ be a primitive $n$-th root of unity. I know that $\text{Gal}(F(\xi)/F)$ is isomorphic ...
0
votes
1answer
38 views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
2
votes
1answer
45 views

Determine the degree of an extension field over $\mathbb{Q}$

Let $\alpha = e^{\frac{i\pi}{6}}$. Compute $[\mathbb{Q}(\alpha):\mathbb{Q}]$ and find the minimal polynomial of $\alpha$, $m_{\mathbb{Q}}(\alpha)$. I can see clearly that $\alpha^6+1=0$ but I ...
4
votes
1answer
41 views

Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb ...
2
votes
2answers
45 views

Confusion about elements in fields, like -1 in Z5

I'm learning field and ring theory, and I've repeatedly seen the usage of -1, -2 and -3 as elements of $\mathbb{Z}_5$. As far as my knowledge goes, $\mathbb{Z}_5$ consists of {0,1,2,3,4}. This is ...
4
votes
1answer
62 views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
0
votes
1answer
40 views

Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
2
votes
1answer
70 views

Extensions of the quadratic closure of $\mathbb{Q}$

I'm looking for extensions of the quadratic closure of $\mathbb{Q}$ of degree $3$, degree $5$. Furthermore, Does there exist an extension of degree $4$? What I know: I know that the ...
0
votes
0answers
17 views

Transcendental field extensions obtained by taking quotient of $k[X_1, \ldots, X_n]$

Given a field $k$, $n \in \mathbb{N}\setminus \lbrace 0 \rbrace$ and $M$ a maximal ideal of $k[X_1, \ldots, X_n]$, can the field $L = k[X_1, \ldots, X_n]/M$ ever be transcendental over $k$? By this I ...
2
votes
1answer
24 views

Which cyclotomic fields are different?

For $n$ a positive integer, let us write $\zeta_n = e^\frac{2 \pi i}{n}$, a primitive $n$th root of unity. It is clear that, if $m$ divides $n$, then we have an inclusion of cyclotomic fields $$ ...
0
votes
1answer
15 views

Systematic way of expressing field extensions

If a field $Q$ were to be extended to include roots of the quadratic polynomial $x^2$$-2=0$, the extended field $Q$($\sqrt2$) would include elements of the form $a$ + $b$$\sqrt 2$. However, extending ...
0
votes
1answer
30 views

Determining automorphisms of this extension

I'm having a bit of trouble understanding an example from the introduction to galois theory section from Gallian's text. We have that $\omega = -1/2 + i\sqrt 3/2$ satisfies the equations $\omega^3 = ...
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0answers
37 views

does isomprphic fields have exactly the same properties?

It is written in many books that isomorphic fields have exactly the same properties. Does that mean only to the algebraic properties (i.e. properties that derived from the field operations)? To ...
0
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0answers
10 views

When does a system of n symmetric polynomials in n variables have exactly one solution over C up to permutation?

I was slightly amused that if I never learned about polynomials and was asked if Vieta's system of equations has exactly one solution up to permutation, the solution would be to develop polynomials in ...
0
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3answers
27 views

What are the ideals of $F_2[x]/\langle x^2 + x +1\rangle$? [closed]

Is it just the divisors of $x^2 +x+1$ in mod $2$ ?
2
votes
2answers
74 views

The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
0
votes
3answers
25 views

Ring Extension: Mapping: $ \mathbb Q[\sqrt d] \rightarrow \mathbb Q$

Show that the Norm: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q, (r+s\sqrt d) (r-s\sqrt d) = r^2 - ds^2$ is multiplicative, i.d. $N(xy) = N(x)N(y)$ How to show it without computing? (I tried to do it ...
0
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1answer
31 views

A problem about degrees of minimal polynomials for two arbitrary elements in an extension field

I'm struggling to come up with a reasonable proof for the following problem: Suppose $E$ is an extension field of a field $K$ and that $a$ and $b$ are algebraic elements in $E$. Show that the ...
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2answers
39 views

Extension fields, and their cardinality and roots

I have no idea how to begin answering this question. My notes do not help. Let $f(X) = X^3$ + $X + 1 ∈ \mathbb Z_5[X]$ ($\mathbb Z_5$ denotes the integers mod 5). Let $E=\mathbb Z_5[X]/(f(X))$. ...
1
vote
1answer
31 views

Show that $f$ is the minimal polynomial of $u$

Let $u$ be a root of $f=x^3-x^2+x+2\in \mathbb{Q}[x]$ and $K=\mathbb{Q}(u)$. Prove that $f=m_\mathbb{Q}(u)$. I have no idea how to approach this problem. Should I prove that $f$ is irreducible ...
0
votes
2answers
39 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated ...
0
votes
1answer
16 views

Unclear explanation of solution again;field extension

The solution sheet assumes additional knowledge than what is provided, which annoys me; I don't understand this. Here's the problem $L:K$ is a field extension. If $\alpha,\beta \in L$ is ...
1
vote
1answer
20 views

why aren't finite fields of prime characteristic algebraically closed?

How can this be proven? I know that if a field has a prime characteristic, any element of the field, say $a$. will satisfy the following equation: $ap = 0$, where p is the prime characteristic of ...
1
vote
0answers
33 views

Since field extensions are vector spaces over ground fields, how is linear algebra used to study them?

I've already taken a semester of Galois theory, so I'm not asking about anything that would be covered there, like the tower theorem. Specifically, multiplication by an element of a field extension is ...