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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Is there a more efficient way of proving a polynomial is primitive?

I have found an impressive Java implementation of a $ℤ2$ Polynomial class in which there seems to be an efficient implementation of a test for reducibility (see ...
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66 views

Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
3
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243 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
3
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157 views

(Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
3
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136 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
3
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119 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
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94 views

The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$

I'm trying to solve this question: Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a field $F$. Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the ...
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144 views

When the extensions Q(α,β) and Q(αβ) over Q are the same?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha\beta)$ over $\mathbb{Q}$ are the same. Thanks in advance.
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909 views

Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals ...
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389 views

field embedding

I've come across this problem in Etingof's notes on representation theory (Problem 5.1 on p. 78). It just sounds nice exercise... The question is : Let $f : k(x_1,\ldots,x_n)\rightarrow ...
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25 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 ...
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66 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 ...
2
votes
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39 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
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19 views

Automorphisms of field generated by two coprime elements

I would like to know if the follwing statement is true: Let $F$ be a field and let $a,b$ be algebraic over $F$ with coprime degrees $m$ and $n$, respectively. Suppose $F(b)/F$ is normal. Letting $K = ...
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20 views

On the restriction of field homomorphisms to subfields

Let $F/L/K$ be field extensions with $L/K$ finite. Let $H=\text{Hom}_K(L,F)$ be the set of field homomorphisms $L\rightarrow F$ that fix $K$. Take $\alpha\in L$, and let $\lbrace ...
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21 views

Quick help on why this extension is of degree $2$

The set up is, Splitting field $K=\mathbb{Q}(i,\alpha)$ where $\alpha=2^{\frac{1}{4}} \in \mathbb{R}$. The three obvious subfields of $K$ with degree $2$ over $\mathbb{Q}$ are..? Answer is ...
2
votes
0answers
21 views

Maximum ideal in field

Let $k$ be a field, $n ∈ \mathbb{Z}>0$, and $α_1, α_2, ..., α_n ∈ k$. Prove: $(x_1 − α_1, x_2 − α_2, \ldots, x_n − α_n)$ is a maximal ideal. I cannot figure out how to prove this; what is meant ...
2
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38 views

Proof that $\mathrm{Aut}(K\left[ \sqrt[m]{g \in K} \right])/\mathrm{Aut}(K) = \mathbb{Z}_w, w | g $

Let $K$ be an extension field of $\mathbb{Q}$. That is $$ K = \mathbb{Q}[r_1,\ldots,r_k ]$$ I am considering $\mathrm{Aut}(K)$ which is the group of field automorphisms of $K$ and I wish to show ...
2
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21 views

Transcendental extension not isomorphic to its closure

Suppose I'm given a field extension $K/F$ with $\alpha\in K$ transcendental over $F$, the claim is that $F(x)\cong F(\alpha)$. It's a statement without proof in our class notes, and the remarks ...
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62 views

What is the number of subgroups of $C_2 \times C_2 \times C_2 \times \cdots \times C_2$?

I think that counting the number of subgroups of various groups is usually very difficult. I was wondering about the number of subgroups of $(C_2)^n$. For example, there are 5 subgroups of $C_2 ...
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47 views

Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the ...
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49 views

I don't understand what a Pythagorean closure of $\mathbb{Q}$ is; how are these definitions equivalent?

I have two definitions of the said field. And frankly I don't see why one is equivalent to the other. It just doesn't add up. Let's look at wikipedia's definition. In algebra, a Pythagorean field ...
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48 views

Proving that primes of the form $4k + 1$ are not Gaussian primes

Let $p$ be a positive integer prime such that $p\equiv1\pmod4$. Assume that $p$ is a Gaussian prime, so $(p)$ is a maximal ideal of $Z[i]$, and consider the field $K = Z[i]/(p)$. We state without ...
2
votes
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99 views

Number of homomorphisms from a stem field to a given field

This is a homework, but I've generalized it as possible in order not to have exact answer rather that to understand the very principle of solution. The problem is following, for a given irreducible ...
2
votes
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47 views

Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub ...
2
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0answers
46 views

Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
2
votes
0answers
28 views

How algebraically restrictive is the structure of $\Bbb Q_p$?

My question is a little difficult to explain, but I will try to state it. First, let me talk about the field $\Bbb R$ of real numbers. Let $K$ be the maximal real algebraic extension of $\Bbb Q$, that ...
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22 views

Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
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83 views

How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$

I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt ...
2
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39 views

Show that $|\text{Hom}_k(K,\widetilde{K})(\phi)|\le [K:L]$

Let $K/k$ be a finite field extension, $L$ an intermediate field and $\widetilde{K}$ such that $\widetilde{K}/k$ is normal. Let $\phi \in \text{Hom}_k(L,\widetilde{K}) := \{\psi:L\rightarrow ...
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36 views

Galois extension of $\mathbb C$ is also Galois over $\mathbb R$??

Is any Galois extension of $\mathbb C$ also Galois over $\mathbb R$? I know if that extension is finite, Then it is true because $\mathbb C$ is algebraically closed. But How about the infinite case? ...
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votes
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102 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) = ...
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0answers
31 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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37 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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43 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
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46 views

Connection between Algebraic structures and Formal Systems

I am a college student with very little mathematics background (up to Calculus 152 at Rutgers University), but have become increasingly interested in computing and mathematics in the last year. I am ...
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12 views

Show that $K^\sigma $ is a field, and some other properties.

Let $K$ a field and $\sigma \in Aut(K)$ (the set of automorphism $K\to K$). Let denote $K^\sigma =\{x\in K\mid \sigma (x)=x\}$ the set of fix point. 1) Show that $K^\sigma $ is a subfield of $K$. ...
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60 views

Give an example of a finite extension of fields that is neither separable nor normal.

The title says it all. This is not homework. Haven't made much progress - I know that with $K=F_2(t)$, $L=K(\sqrt t)$ (i.e. adjoining the square root of $t$ to the function field with coefficients in ...
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42 views

Determine All Divisors of $f(x)=x^n\in F[x]$

Carefully determine all divisors of $f(x)$ where $$ f(x)=x^n\in F[x]$$ note that $F[x]$ is a Field So, $$ \underbrace{x^0\mid x^n,\ x^1\mid x^n,\dots,\ x^n\mid x^n}_{n+1}$$ making $n+1$ divisors. ...
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39 views

Squares of $F_{467}$ and 7 as a quadratic residue of a prime mod that prime

I am having a hard time understanding what exactly is meant by this question. Could someone give me a solution with a clear explanation? If $x=467$, are 111 127 and 225 squares in $F_x$? Explain your ...
2
votes
0answers
40 views

Let $K=\Bbb Q(X)$ where $X=\{\sqrt p: p$ is prime$\}$. How to conclude that $|Gal(K/\Bbb Q)|>[K:\Bbb Q]$

Let $K=\Bbb Q(X)$ where $X=\{\sqrt p: p$ is prime$\}$. Then $K$ is galois over $\Bbb Q$. If $\sigma \in Gal(K/ \Bbb Q)$, let $Y_{\sigma}=\{\sqrt p: \sigma(\sqrt p)=-\sqrt p\}$. Then how to prove a) ...
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35 views

Is the symmetric group $S_{\mathbb{N}}$ on countably many letters a Galois group?

We know $S_n$ for $n \in \mathbb{N}$ is the Galois group over some field $K$ (in fact also over $\mathbb{Q})$. The construction of field $K$ and extension $L/K$ such that ...
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26 views

When is a gradient vector field also an algebraic field?

I was thinking about the inconsistencies in mathematical vocabulary today, and I came to this simple, open-ended question: When is a vector field (or a gradient field) an honest algebraic field? ...
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38 views

Maximal cyclic subextension of Galois extension

Let $K$ be a Galois number field with cyclic Galois group. Let $L$ be an abelian Galois number field such that $K\subseteq L$. Suppose that there are no Galois subextensions $\mathbb Q\subseteq F ...
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67 views

Cancellation with scalars in vector space, a useless law?

In typical textbooks on vector spaces the axioms are stated and then several algebraic identities are proven. Among them are the cancellation properties $$ x + y = x + z \mbox{ implies } y = z $$ and ...
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35 views

Is $2^e$ in the field extension $\mathbb{Q}(e)$?

As the title says, is $2^e$ in the field $\mathbb{Q}(e)$? I mostly study analysis, but this came up trying to answer someone else's question. So far, my idea has been to suppose it's true and use the ...
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0answers
39 views

Show that $K=k(u)$ iff $u=\frac{ax+b}{cx+d}$ where $a,b,c,d \in k$ with $ad-bc \neq 0$

Let $k$ be a field and let $K=k(x)$ be the rational function field in one variable over $k$. If $u\in K$, show that $K=k(u)$ iff $u=\frac{ax+b}{cx+d}$ where $a,b,c,d \in k$ with $ad-bc \neq 0$. ...
2
votes
0answers
24 views

Proving that the set of separable elements over a field is a field itself.

My field theory book says that the set of separable elements over a field is a field itself. This roightly translates to the fact that of $a $ and $b $ are separable, so are $a+b, ab, 1/a$, etc. I ...
2
votes
0answers
76 views

Proving that $t^{p^r}-a$ is irreducible when $a\in k$ is not a $p$th power

Let $p$ be an odd prime, $F$ a field of characteristic $0$ and $a\in F$ with $a\neq 0$. Assume $a$ is not a $p$th power in $F$. Prove that for every positive integer $r$, $t^{p^r}-a$ is irreducible ...
2
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59 views

A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of ...