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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4
votes
1answer
37 views

$|\chi(\mathfrak{a})| = 1$ for any ideal $\mathfrak{a}$?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
1
vote
1answer
34 views

Understanding Abel-Ruffini

I'm wondering of anyone can point me towards a proof of why we can't have a quintic formula, using concepts from basic group theory. In particular, I understand that there is some connection with ...
-1
votes
1answer
39 views

In finite field why $\overline c = -c \ne 0$ for $\overline c = c^{p^f}$.

Let $GF(p^{2f})$ be a finite field of order $p^{2f}$. Consider the map $\overline x := x^{p^f}$ for $x \in GF(p^{2f})$. Let $b \in GF(p^{2f}) - GF(p^f)$ and set $c := b - \overline b$. Why do we ...
0
votes
1answer
20 views

When is $u_1 = w_1 + aw_2, u_2 = w_2 + a w_1$ a basis if $w_1, w_2$ is a basis.

If $V$ is a $2$-dimensional $K$-vector space with basis $w_1, w_2$, when is $$ u_1 := w_1 + a w_2 \qquad u_2 := w_2 + a w_1 $$ is basis. Of course for $a = 1$ it is certainly not, but how could ...
3
votes
0answers
28 views

The degree of a primitive element

Suppose you have a map of smooth projective irreducible curves $X \to Y$ over $\mathbb{k}$ of degree $r$. This give an extension of fields $\mathbb{k}(Y) \hookrightarrow \mathbb{k}(X)$. The primitive ...
0
votes
1answer
18 views

Difference/similarities: Field of rational functions in X - smallest field containing K and x?

I am a bit confused with the Notation $K(X)$ or $K(x)$. In my notes I found two definitions: $K(X)$ is the field of rational functions with indeterminate $X$. $K(x)$ is the smallest field ...
2
votes
0answers
56 views

Is the splitting field of $g=x^3-3x-1$ over $\Bbb Q$ a radical extension?

Since $g=x^3-3x-1$ is irreducible over $\Bbb Q$, and has square discriminant. If $L$ is the splitting field of $g$ over $\Bbb Q$, since $g$ has square discriminant we have $\text{Gal}(L/\Bbb Q) = ...
0
votes
1answer
56 views

Find an $x$ such that $\mathbb{Q}(x) = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$.

Find an $x$ such that $\mathbb{Q}(x) = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$. For my abstract algebra class. Don't really know where to start, or how to finish for that matter.
12
votes
1answer
149 views

Elliptic curve over algebraically closed field of characteristic $0$ has a non-torsion point

Let $E/k$ be an elliptic curve over an algebraically closed field $k$ of characteristic $0$. Can one prove that the abelian group $E(k)$ is non-torsion? Better yet, can one prove that $E(k) ...
2
votes
2answers
35 views

Minimal polynomial of integral elements [duplicate]

Let $R$ be an integrally closed domain and let $K$ be its fraction field. Let $L\supseteq K$ be a field. If $\alpha\in L$ is integral over $R$ (i.e. if it satisfies a monic polynomial in ...
3
votes
1answer
36 views

The projective special linear group $PSL(2,\mathbb F)$ acts equivalent on $\mathbb F^2$ and $\mathbb F_{\infty} = \mathbb F \cup \{\infty\}$

This question is closely related to this one. Again consider the group $G := PSL(2, \mathbb F)$ over some field $\mathbb F$. Then as written in the other post, there are two natural actions ...
0
votes
1answer
38 views

Finite separable extension of a field, free module.

I'm trying to understand proof of following theorem. $\textbf{Theorem.}$ If $L$ is finite and separable extension of field $K$, $K$ is field of fractions of principal ideal domain $A$ and $B$ is ...
0
votes
1answer
25 views

Example for an algebraic field extension

Let $\mathbb{F}_p$ be the field with $p$ elements. Let $L:=Quot(\mathbb{F}_p[t])$ and $f\in L[x]$ be irreducible. Let $K:=L[x]/(f)$ and $w:=x+(f)\in K$. Consider $\mathbb{F}_p(w)\subset K$. Problem: ...
2
votes
1answer
26 views

Prove that $K$ is algebraically closed

Let $K$ be a field of characteristic zero and $p$ a prime number such that $p^2$ divide at degree of all irreducible polynomial not linear in $K[x]$. Prove that $K$ is algebraically closed. Thank you ...
1
vote
0answers
43 views

On the relation between $PSL(2,\mathbb F)$ and $PGL(2,\mathbb F)$.

On wikipedia:Möbius transformation in the linked paragraph it is shown that $$ PGL(2, \mathbb C) \cong \mbox{Aut}(\hat{\mathbb C}) \quad \mbox{and} \quad PSL(2, \mathbb C) \cong ...
2
votes
1answer
76 views

How to compute in the multiplicative group of finite field “economically” and efficiently

Let $GF(p^n)$ be a finite field, then the additive group is isomorphic to $\mathbb Z / (p\mathbb Z)^n$, and it is simple to compute in that group. The multiplicative group is always cyclic (a standard ...
1
vote
1answer
30 views

Unicity of the copy of $\mathbb{Q}$ in an ordered field.

One can prove that every ordered field contains a copy of the ordered field $\mathbb{Q}$. To show this, one can first show that every ordered field $K$ must be of characteristic $0$, and use this ...
2
votes
0answers
27 views

Showing that $|\text{Hom}_\mathbb{Q}(K,K)|=6$

Let $g\in \mathbb{Q}[X]$ be irreducible with $\text{deg}\;g=3$. We assume that only one of the roots of $g$ is in $\mathbb{R}$. So, $\alpha \in \mathbb{R}$. Let $L\subset\mathbb{C}$ be the splitting ...
2
votes
1answer
88 views

How can we prove $\mathbb{Q}(\sqrt 2, \sqrt 3, … , \sqrt n ) = \mathbb{Q}(\sqrt 2 + \sqrt 3 + … + \sqrt n )$

I want to prove this statement. $$\mathbb{Q}(\sqrt 2, \sqrt 3, ..... , \sqrt n ) = \mathbb{Q}(\sqrt 2 + \sqrt 3 + .... + \sqrt n )$$ for any $n >1$. It looks like a very hard problem. How ...
0
votes
2answers
33 views

Intermediate Galois fields

I want to find two different fields $K_1,K_2$ such that $\Bbb Q\subset K_i \subset \Bbb Q(\alpha,\zeta)$ such that $K_i /\Bbb Q$ are Galois. A few things: $\alpha$ is the real $6^{th}$ roof of $2$, ...
2
votes
1answer
40 views

Galois group of $x^6-2$ over $\Bbb Q$

I want to find the Galois group of $x^6-2$ over $\Bbb Q$. I have posted my attempt in an answer below. Is there a better way? Alternative proofs are greatly appreciated(especially shorter ones ...
5
votes
2answers
62 views

What is a constant field?

I am looking at the following: Could you explain to me what a constant field is? $$$$ P.S. I found this in the paper of T. Honda, "Algebraic differential equation" (pages 170-176).
0
votes
1answer
41 views

Explicit basis for normal closure of $\Bbb Q(\alpha)$ over $\Bbb Q$

Let $\alpha$ denote the real $6^{th}$ root of $2$, and let $L$ denote the normal closure of $\Bbb Q(\alpha)$ over $\Bbb Q$. I want to give an explicit basis for $L$ as a $\Bbb Q$-vectorspace. In ...
1
vote
0answers
21 views

Definition of a radical extension? Correct interpretation?

I have here defined: $L/K$ is a radical extension iff there is a tower of fields $K=K_0\subseteq K_1\subseteq \cdots \subseteq K_m = L$ with $K_{i+1}=K_i(\alpha_i)$ and $\alpha_i^{n_i} \in K_i$ ...
3
votes
2answers
49 views

Degree of splitting field for $x^3-3x-1$ over $\Bbb Q$ and $\Bbb F_5$

I want to find the degree of the splitting field for $x^3-3x-1$ over $\Bbb Q$ and $\Bbb F_5$. My attempt is contained below.
0
votes
0answers
29 views

An exercise about repeated root

I want to prove the following: Let $K$ be a field, let $g(X) \in K[X]$ be an irreducible polynomial, and let $L$ be a splitting field for $g(X)$ over $K$. Prove that $g(X)$ has at least one ...
1
vote
1answer
39 views

Are $A=\Bbb Q(\sqrt{2}),B=\Bbb Q(i),C=\Bbb Q(i\sqrt2),D=\Bbb Q(i,\sqrt2)$ normal over $\Bbb Q$?

Are $A=\Bbb Q(\sqrt{2}),B=\Bbb Q(i),C=\Bbb Q(i\sqrt2),D=\Bbb Q(i,\sqrt2)$ Normal over $\Bbb Q$? My attempt is in the answer below. Is this theorem stated correctly? Are there hidden obstructions ...
23
votes
4answers
1k views

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of ...
2
votes
0answers
35 views

Prove that $F$ is finite field

We denote $\mathbb{F}_q$ as finite field of $q$ elements and the algebraic clausure as $\overline{\mathbb{F}}_q$. Let $$K=\bigcup_{n=1}^{\infty}\mathbb{F}_{q_n}\subset \overline{\mathbb{F}}_3,\; ...
1
vote
1answer
47 views

$\mathbb{C^2}$ treated as a real vector space

As a real vector space the dimension is 4. What is an orthonormal basis for it with respect to either the standard real inner product?. I've tried gram schmidt with the obvious choice of 4 vectors( ...
2
votes
1answer
51 views

Uniqueness of the real line

A few days ago, I came across this question in a review queue. I tried my luck at it. Here is what I did: If I want a homomorphism (isomorphism, but even just homomorphism) $f:\mathbb{R}\to F$, then ...
1
vote
2answers
51 views

Constructing a field so that a polynomial has a root.

This was on my study guide given out by my professor. I could not find this anywhere in my book though. Any advice on how to do this. I understand the concept that a polynomial may not have a root ...
0
votes
2answers
36 views

The minimal polynomial of a root depends on the field it belongs to

The minimal polynomial of a root is dependent on the field it lies in. Correct? I was thinking that to find the minimal polynomial of $\sqrt{2}$ over $\Bbb Q$ meant to do the following: $$\alpha ...
0
votes
0answers
20 views

Existence of Galois extension [duplicate]

Let $G$ be a finite group. Then there exist fields $L$ and $K$ such that $L$ is an extension of $K$ with Galois group $G$. I think that since $G$ is finite, then $G$ must isomorphic to a subgroup of ...
3
votes
2answers
131 views

Why does Rudin require $1\neq 0$ in order for a set to form a field? [duplicate]

The following comes from Baby Rudin: Is there any special reason as to why Rudin makes $1\neq 0$ a requirement? I've seen he uses this fact in a few elementary proofs below, the only ...
2
votes
1answer
34 views

Generalizing the norm and trace of finite extensions over finite fields.

I'm currently reading through Ireland and Rosen's A Classical Introduction to Modern Number Theory, and I'm working on proving that a later definition of trace and norm of arbitrary finite algebraic ...
0
votes
1answer
35 views

If two fields are isomorphic then so are their algebraic closures?

If $E,F$ are isomorphic fields, is it always true that $\bar E\cong \bar F$? Now suppose $f:E\to F$ is an isomorphism, I want to construct an isomorphism $f':\bar E\to \bar F$ based on $f$. If this ...
0
votes
1answer
37 views

Prove or Disprove $I=\{a_0x^0+\dots +a_nx^n:a_0+\dots+a_n=0\}$ is a sub ring of field F[x]

Prove or Disprove $I=\{a_0x^0+\dots +a_nx^n:a_0+\dots+a_n=0\}$ is subring of field F[x] We need to show it is a subring that is closed addition closed multiplication (struggling) $\exists $ ...
1
vote
1answer
16 views

Quadratic extensions - understanding

If $[L:K]=2$ then $L/K$ is a quadratic extension. Can we think of a quadratic extension as $L=K(\alpha)$ where $\alpha$ is anything not in $K$(be it algebraic or transcendental)? This makes $L$ a ...
2
votes
1answer
109 views

Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$

For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$? This problem appears to be quite common, however my understanding of Galois ...
2
votes
0answers
37 views

To prove a equality of field norm by field extension.

We know that if we set $K$, $F$ and $L$ fields, with $L$ a finite extension of $F$ and $F$ a finite extension of $K$. Then we have the norm equality $N_{L/K}=N_{F/K}N_{L/F}$. The common solution is by ...
2
votes
1answer
34 views

Field Extension of degree $2$ is Normal

I am trying to prove that if a field extension $E$ over $F$ is such that $$ \left[ E : F \right] = 2, $$ then $ E $ is a normal extension over $F$. My approach to solve this is take an element $ a ...
1
vote
1answer
61 views

Showing that a ring is a field as well for one of the provided choices.

Let $\mathbb M$ be one of the following rings: $\mathbb{R}, \mathbb{Q}, \mathbb{F_9}, \mathbb{C} $. Let $I$ be the ideal generated by $x^4+2x-2$. Is the ring $\mathbb M[x]/I$ a field for some of ...
3
votes
2answers
34 views

$K$-homomorphisms from an étale $K$-algebra to a field

I am somewhat confused by the proof of Theorem 1.5.4 (p.22) (Grothendieck's version of the main theorem of Galois Theory) in Szamuely's Galois Groups and Fundamental Groups. The theorem establishes an ...
5
votes
2answers
87 views

Explain how to compute $\cos(2\pi/13)$ by solving quadratic and cubic equations only

I know that we can express $2\pi/13$ as a root of unity on the unit circle taking $z^{13} = 1$ and $z=\cos(2\pi/13)+i\sin(2\pi/13)$ and that we should be able to find a polynomial for this over the ...
2
votes
1answer
51 views

Is $e^{x}$ algebraic over $\mathbb{C}(x)$?

We have $$e^{x}= \sum \frac{x^{n}}{n!}$$ as a power series expansion. We can see that such a power series has radius of convergence infinity and as a result $e^x$ is defined over the whole complex ...
2
votes
2answers
39 views

Elements of $\mathbb{C}(x)$ and algebraic elements over $\mathbb{C}(x)$

Well, the elements of the ring $\mathbb{C}[x]$ are easy to understand for me. They can be thought of as polynomial functions from $\mathbb{C} \rightarrow \mathbb{C}$ and as a result they are ...
-2
votes
1answer
34 views

Irreducible polynomial proof.

Pretty lost on how to go about this. Think I'm missing some facts or theorems.
4
votes
2answers
62 views

Solving an $x^2 +x-1$ in a field with $49$ elements

I was given a problem that I don't seem to know how to solve. It says Let $\mathbb{F}$ be a field with $7$ elements. Construct a field $\mathbb{L}$ with $7^2$ elements and show that $x^2-3$ and ...
0
votes
3answers
52 views

Isomorphism of fields.

Find an isomorphism from the quotient field $\mathbb{Q}[x]/(x^2-2)$ to the field $\mathbb{Q}[\sqrt{2}]=\{a+b\sqrt{2}\ |\ a,b\in\mathbb{Q}\}$ Having a lot of trouble figuring this out.