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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
32 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 $2$, ...
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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 ...
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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).
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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 ...
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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 $\alpha_i^{n_i} \in K_i$ ...
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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.
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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 ...
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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 ...
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4answers
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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 ...
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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,\; ...
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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( ...
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1answer
50 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $ ...
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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
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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
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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 ...
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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 ...
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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
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2answers
33 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 ...
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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 ...
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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 ...
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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 ...
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1answer
34 views

Irreducible polynomial proof.

Pretty lost on how to go about this. Think I'm missing some facts or theorems.
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2answers
61 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 ...
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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.
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2answers
54 views

Factoring $x^3-3x-1$ in terms of $\alpha$ unknown

I never got a satisfactory answer here: Factoring $x^3-3x-1\in \Bbb Q[x]$ in terms of a unknown root But all context is contained below I want to factor $x^3-3x-1\in \Bbb Q[x]$ in terms of ...
0
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2answers
44 views

Is it true that is $P^n=\langle a^n\rangle$ such that n is a positive integer?

Let $R$ be principal ideal domain that is a local. Let $P$ be an ideal of a ring $R$. Let $P$ be generated by an element $a$ (i.e. $P=\langle a\rangle$). Is it true that is $P^n=\langle a^n\rangle$ ...
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1answer
49 views

Find the automorphisms for the Galois Group of the minimial polynomial $x^4+1$.

Determine the splitting field $L$ for this polynomial over $\mathbb{Q}$. The splitting field of $x^4+1$ must contain the solutions to $$x^4+1=0,$$ that is, $x^4=-1$. So $x^2=\pm i$, and ...
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2answers
50 views

Why $Gal(\mathbb Q(\zeta _n)/\mathbb Q)\hookrightarrow (\mathbb Z/n\mathbb Z)^\times $

Let $\zeta _n=e^{\frac{2i\pi}{n}}$ an let $\mu_n=\{1,\zeta _n,\zeta _n^2,...,\zeta _n^{n-1}\}$. 1) Show that $\mathbb Q(\mu_n)=\mathbb Q(\zeta _n)$ and that $\mathbb Q(\zeta _n)/\mathbb Q$ is ...
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0answers
28 views

Is this the correct basis of $K(\alpha,\beta)$

Basis of $K(\alpha,\beta)$ So $K(\alpha,\beta)=K(\alpha)(\beta)=K(\beta)(\alpha)$, and hence we can form: $K(\alpha)$ is formed by taking $K[x]/\langle f\rangle$ where $f$ is the minimal polynomial ...
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22 views

show $\alpha=t^{1/p}$ is not separable over $F.$

I had this question in my exam ,but don't know how to proceed: Let $F=\mathbb Z/p\mathbb Z(t)$ where $t$ is indeterminate. Then show $\alpha=t^{1/p}$ is not separable over $F.$ What I know is ...
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2answers
35 views

Find all irreducible polynomials of degrees 1,2 and 4 over $\mathbb{F_2}$.

Find all irreducible polynomials of degrees 1,2 and 4 over $\mathbb{F_2}$. Attempt: Suppose $f(x) = x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \in \mathbb{F_2}[x]$. Then since $\mathbb{F_2} =${$0,1$}, then ...
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2answers
29 views

Show Galois extension with $[K : F] = n$ has an intermediate filed with $L$ with $[K : L] = p$, where $p$ is a prime divisor of $n$

It is a two part question. Let $K/F$ be a Galois extension with $[K : F] = n$. If $p$ is a prime divisor of $n$, prove there is an intermediate field $L$ with $[K : L] = p$. Prove or disprove that ...
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1answer
33 views

Why $\mathbb F_q/\mathbb F_p$ where $q=p^n$ is an extension of degree $n$ ? compute his Galois group.

Let $q=p^n$ where $p$ is prime. Q1) Why is $X^{q}-X\in \mathbb F_p[X]$ irreducible ? Don't we have $X^q-X=X(X^{q-1}-1)$ ? Q2) Suppose it is irreducible, his degree is $p^n$, then $[\mathbb ...
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2answers
27 views

Determining whether the quotient $\mathbb{Q}[x] / \langle f(x) \rangle$ of a polynomial ring over a field is itself a field

I want to know whether $$\mathbb{Q}[x]/(x^3-1)$$ is a field or not. Is it as simple as determining if $x^3-1$ is irreducible in $\Bbb Q$? But since it has roots , $x=1$ wouldn't this imply it is ...
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0answers
51 views

Field extension is étale implies polynomial is separable

Following Johnstone (Exercise 0.11), a ring homomorphism $f: A\rightarrow B$ is étale if for every nilpotent ideal $N\subseteq R$ of a ring $R$ and every diagram of ring homomorphisms there is a ...
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0answers
43 views

Product of elements in finite field

Let $q$ be a prime or primer power such that $\mathbb{F}_q$ is a finite field. Now consider the extension $\mathbb{F}_{q^m}$, which may be regarded as a vector space of dimension $m$ over ...
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25 views

Change of Degree of a Field Extension after adjoining a new element

I'm working on a question saying $A \subseteq B \subseteq C$ fields, and $a \in C$ algebraic over $A$, either prove $[A(a):A] ≥ [B(a):B]$ or give a counter example. I think it's true and used the ...
3
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0answers
32 views

Is there a general algebraic notion of the chain rule?

To motivate this, I should explain that I have been studying differential fields, i.e. fields endowed with a differentiation operator such that $(a+b)'=a'+b'$ and $(ab)'=a'b+ab'$. Using these rules, ...