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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1answer
18 views

Hilbert's Nullstellensatz - question about generalization

It is well-known that if $k$ is algebraically closed field that has infinite transcendence degree over the prime field $\mathbb{Q}$ or $\mathbb{F}_p$ then the maximal ideals of $k[x_1,...,x_n]$ are of ...
0
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1answer
29 views

What is the smallest $m>0$ for the Frobenius automorhism of a Galois Field to be the identity?

Surprisingly scarce information on this particular problem. $\phi$ is the Frobenius automorphism of $GF(p^n)$ for some prime $p$. Find the smallest $m>0$ such that $\phi^m$ is the identity ...
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0answers
21 views

Finite Groups as Galois Groups of Field Extensions [duplicate]

Let $F$ be a field, and $x_{1}, \dots, x_{n}$ free variables. Let $K=F(x_{1},x_{2},\dots,x_{n})$, being the fraction field of the ring of polynomials in the variables $x_{1},\dots,x_{n}$. Show that ...
1
vote
1answer
22 views

Jordan-Chevalley decomposition for non-algebraically closed fields?

Say that we have a field $\mathbb{K}$ which is not necessarily algebraically closed, and $V$ a finite-dimensional vector space over $\mathbb{K}$. Given an endomorphism $X \in ...
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2answers
36 views

Field extension with odd degree

Let $F(a)$ be Field extension over $F$ such that $[F(a):F]=5$. I know that if $[F(a) : F]$ is odd then $F(a) = F(a^2)$. So how can I show that $F(a)= F(a^2+a+1)$. Could somebody please give me hints. ...
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2answers
34 views

Galois Extension with Galois Group $(\mathbb{Z}/2\mathbb{Z})^{3}$

Write an example of a Galois extension of fields that has as a Galois group $(\mathbb{Z}/2\mathbb{Z})^{3}$ I'm not very familiar with Galois theory, so I don't know of a general procedure to ...
2
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2answers
40 views

Rational Polynomial of Degree $3$ satisfying $2\cos{(2\pi/7)}$

Let $\eta = \zeta_{7}+\bar{\zeta_{7}}$, for $\zeta_{7}=\exp{(2i\pi/7)}$. Find a polynomial of degree 3 with rational coefficients that $\eta$ satisfies. I'm not so sure on how to begin by this ...
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2answers
49 views

What does the terminology “characteristic polynomial” mean?

In the documentation of the command "MininimalPolynomial" in Mathematica it says "Find the characteristic polynomial of Sqrt[2] over the extension E^(I Pi/4). The code: MinimalPolynomial[Sqrt[2], x, ...
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votes
1answer
24 views

$a$ is algebraic over $k(b)$ where $b=g(a)$ for some non constant polynomial $g$ [closed]

Let $k \subset K$ be a field extension and $a \in K$. Show that if $g \in k[x]$ is any nonconstant polynomial and $b = g(a)$, then $a$ is algebraic over $k(b)$.
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2answers
43 views

Is $K[a,b]=K(a,b)$ for algebraic $a, b$?

Consider a field extension $L/K$ and $a,b \in L$ algebraic over K. I know that $K[a]=K(a)$ if and only if $a$ is algebraic over $K$. Is it true that in this setting $K[a,b]$ is already a field?
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1answer
41 views

A polynomial with a root in $\mathbb{F}_p \ \forall p$, where $p$ is prime, but no root in $\mathbb{Z}$ [duplicate]

Give an example of a polynomial $f(x) \in \mathbb{Z}[x]$ which has a root in every finite field $\mathbb{F}_p$, but no root in $\mathbb{Z}$.
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3answers
47 views

Show that $\mathbb{Q}(\sqrt2, \sqrt[3]2)$ is a primitive field extension of $\mathbb{Q}$.

I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare. Thanks in advance
3
votes
1answer
29 views

Intersection of field extensions

Let $F$ be a field and $K$ a field extension of $F$. Suppose $a,b\in K$ are algebraic over $F$ with degrees $m$ and $n$, where $m,n$ are relatively prime. Then $F(a) \cap F(b) = F$. I see that the ...
0
votes
1answer
21 views

Quotient of polynomial ring over an algebraically closed field, isomorphism of fields

Let $K$ be an algebraically closed field. Let $n$ be an integer. Let $M$ be a maximal ideal of the ring of polynomial $K[X_1,...,X_n]$. Then the quotient ring $K[X_1,...,X_n]/M$ a field and a ...
2
votes
2answers
46 views

Why does $F(\sqrt{a+b+2\sqrt{ab}}) = F(\sqrt{a},\sqrt{b})$?

Let $F$ be a field of characteristic $\neq 2$. Let $a \neq b \in F$, and $F(\sqrt{a},\sqrt{b})$ is of degree 4 over $F$. I've shown that $F(\sqrt{a}+\sqrt{b}) = F(\sqrt{a},\sqrt{b})$. Observe that ...
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2answers
23 views

One basis of extension field works for all roots of minimum polynomial?

Let $p(x)$ be a minimum polynomial of $c \in K \supset F$ over $F[x]$. Say $p(x)$ has multiple distinct roots $\{r_i\}$including $c$. I know that $F(c) \cong F(r_i)$ for each $i$, but is a basis of ...
0
votes
1answer
62 views

Existence of $f'\in F$ such that $f^{[G:F]}=(f')^n$ for finite extension $F\subset G$

Let $F$ be a field and $u\in F$. Further, let $F\subset G$ be a finite extension and $v\in G$ such that $v^n=u$ for $n>0$. How would I show that there exists a $\tilde{u}\in F$ such that ...
1
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1answer
45 views

Extension of $\mathbb{Q}$ over which $\pi$ is algebraic of degree 3?

I think $\pi$ is algebraic of degree 3 over $\mathbb{Q}(\pi^3)$. To prove it, I need to show that $\pi \notin \mathbb{Q}(\pi^3)$ (which implies that $x^3-\pi^3$ is the minimum polynomial of $\pi$ over ...
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3answers
89 views

Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$

I'm having trouble finding a basis for $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$. So far I know that $[K=\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2$ and $[L=\mathbb{Q}(\sqrt{2}, \sqrt{3}):K] = 2$, but it's ...
1
vote
1answer
38 views

Show $x^2-2$ is irreducible over $\mathbb{Q}(\sqrt[3]{4})$

I'm having trouble showing that $p(x)=x^2-2$ is irreducible over $\mathbb{Q}(\sqrt[3]{4})$. What I tried: I already know the roots of $p(x)$ are $r=\pm \sqrt{2}$, so I just need to show that ...
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0answers
16 views

$ \tan$ function of two arguments when second part is modified

Can $ \tan( x + n \,y)$ be expressed as a function of $ \tan x, \tan y $ and some $ f(n) ? $ A particular integer validity for $n$ in: $$\tan( x + y \cos n \pi) = \dfrac{\tan x + \ f(n) \tan y ...
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3answers
61 views

How is it true that $-\frac{1}{2}+i\frac{\sqrt{3}}{2} \in \mathbb{Q}(i,\sqrt{3})$?

I am having doubts this is true. So elements of $\mathbb{Q}(i,\sqrt{3})$has the form $p+qi+r\sqrt{3}$ for some rational $p,q,r$. This means that I must be able to express ...
1
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1answer
30 views

If $R$ is an integral domain, show that there is no subfield $k$ of Frac($R$) containing $R$.

If $R$ is an integral domain, show that there is no subfield $k$ of Frac($R$) containing $R$. I think the way to prove this is by contradiction. So, let $R$ be an integral domain, and let $k$ be a ...
3
votes
1answer
61 views

Pythagorean Closure of $\mathbb{Q}$ is extremely obscure. Can someone give a clear answer?

My previous question didn't get me a definite answer. I've refined it so with a specific example so it might help someone clear my confusion. In Stewart's book, The Pythagorean closure is defined as ...
1
vote
1answer
57 views

Show that $\sqrt[15]{63}$ does not belong in $\mathbb Q(\sqrt[189]{147})$ [closed]

How to prove that $\sqrt[15]{63}$ does not belong in $\mathbb Q(\sqrt[189]{147})$?
2
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0answers
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 ...
1
vote
1answer
30 views

Irreducibility over field extension $\mathbb{Q}(\sqrt[3]{2})$

I want to show that $p(x)=x^2-1-\sqrt[3]{2}$ is irreducible over $\mathbb{Q}(\sqrt[3]{2})$. What I tried: Assume by contradiction that $p(x)$ is reducible; then it has a root $r$. We know that $\{1, ...
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4answers
108 views

Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$

For a number field $K$, Dirichlet's unit theorem says that$$(O_K)^\times = \mathbb{Z}^{r - 1} \oplus (\text{a finite cyclic group}),$$where $r$ is the number of all infinite places of $K$. An infinite ...
2
votes
1answer
34 views

What is $\left(\overline{\mathbb{C}(t)}\right)^{\times}$?

In a previous question, I asked about the automorphism groups of the groups of units of various fields. In an answer to this question it is explained that $\mathbb{C}^{\times} \cong S^{1} \oplus ...
2
votes
6answers
103 views

Why is $\mathbb{Q}[\sqrt{2}]$ a field?

It seems to me that the hard part of this proof is multiplicative inverses. I know how to prove this by considering the multiplication of two arbitrary elements: $$(a + b\sqrt{2})(a' + b'\sqrt{2}) = ...
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2answers
26 views

How to prove $\alpha =\{1, x,…,x^n\}$ is a basis for $P_n(F)$

Let $v=a_0+a_1x+...+a_nx^n$, if $a_0+a_1x+...+a_nx^n=0$, then all coefficient $a_i\in F$ are $0$ as variables are all with different degrees so it is linearly independent (Here I am no sure how to ...
0
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1answer
29 views

How to verify that $P_n(F)$ is a vector space over field $F$?

I assume $p(x)=a_nx^n+...+a_0$ where $a_i\in F$ and $x$ is variable. To verify it as vector space, I think we need to check axioms. i.e. $\forall x, y,z\in V,\ (x+y)+z=x+(y+z)$ and $\forall x, y\in ...
2
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2answers
52 views

Field extensions of $\mathbb{Q}$, $\mathbb{Q}(\xi_7)$ and $\mathbb{Q}(\xi_7+\xi_7^{-1})$

Let $\xi_7$ denote the complex number $e^{2\pi i/7}$ and let $\beta = \xi_7+\xi_7^{-1}$, consider the field extensions $\mathbb{Q} \subset \mathbb{Q}(\beta) \subset \mathbb{Q}(\xi_7) $. Determine ...
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0answers
23 views

Explicit way of finding particular field elements

Suppose I am working with the field $\mathbb{F}_{5^2}$ and I know that there exists a primitive element $\gamma$ that satisfies the following equation $\gamma^2 = \gamma + 3$. I want to calculate ...
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2answers
102 views

Find minimal polynomial of $ e^{2\pi i/5}$ over $\mathbb{Q}$

The complex number $z = e^{2\pi i/5}$ is a fifth root of the unity: $z^5 = 1$. Find the minimal polynomial of $z$ over $\mathbb{Q}$. I tried to solve this by converting the z into the term of $a ...
3
votes
1answer
23 views

Invertible elements of a power series ring

Let $F$ be a field and $F[[x]]$ be the power series ring with coefficients in $F$. It seems if $\alpha, \beta \in F[[x]]$, with $\alpha^{-1} = \beta$, then all coefficients of the product $\alpha * ...
2
votes
1answer
74 views

Dimension of the algebraic closure of a continuum field of characteristic zero

Let me start by saying that I have no idea in algebra/number theory/whatever, so, please, forgive my ignorance. Let $\mathbb{F}$ be a field of characteristic zero and continuum cardinality, which ...
2
votes
1answer
76 views

Proving existence of an element of trace 1

Let $F=\mathbb{F}_{q}$ be a finite field of order $q=2^{n}$ and let $\beta$ be a primitive element of $F$. I would like to prove that if $q>4$, then for each $1\leq i \leq \frac{q-2}{2}$, there ...
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1answer
42 views

Prove $F=K(u)$ if $F$ is a cyclic extension of $K$. (Hungerford, exercise V.7.5) [closed]

Prove that if $F$ is a cyclic extension of $K$ of degree $p^n$ ($p$ prime) and $L$ is an intermediate field such that $F=L(u)$ and $L$ is cyclic over $K$ of degree $p^{n-1}$, then $F=K(u)$
3
votes
2answers
56 views

How many different field homomorphisms can we have between any two given fields?

How many field homomorphisms can we have between any two given fields? That is, if $A$ and $B$ are fields, what can we say about the cardinality of $\text{Hom}(A,B)$? I don't know much abstract ...
3
votes
1answer
39 views

Action on its generators of splitting field of $x^4 +5$

Splitting field is $K=\mathbb Q (\sqrt[4]{-5}, i)$ Degree of $K$ over rationals is $8$ so the galois group $G=\text{G}(K/ \mathbb Q)$ has order $8$. $x^4 +5$ is irreducible so there is one orbit ...
3
votes
1answer
55 views

Algebraic numbers in real closed fields

I have been looking at the discussion of real closed fields in Appendix B of Marker's Model Theory:an Introduction. I am baffled by what it says about the uniqueness of real closures. I have no ...
3
votes
1answer
104 views

Intersection between two integral closures equals an algebraically closed field

Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Prove that $A ...
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votes
2answers
32 views

Galois closure of a finite extension is finite

We proved the fundamental theorem of algebra in my field theory class the other day, but the professor glossed over an (imo) important step which I have found myself unable to prove. Can someone help ...
0
votes
1answer
23 views

Please help clarify notation in proof of Kronecker's Theorem of Field Extensions

I'm a little bit confused by the notation being used in a proof of Kronecker's Theorem in Gallian, and I was wondering if you could help me understand what is going on. Theorem: We want to show that ...
1
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0answers
48 views

What are the subfields of $Q(\zeta_7)$

Let $\zeta = e^{\frac{2i\pi}{7}}$. I know that the automorphisms of $Q(\zeta)$ are isomorphic to the cyclic group with $6$ elements so that the subfields of $Q(\zeta)$ correspond to the subgroups of ...
1
vote
1answer
46 views

Find the splitting field without the explicit roots

Let $f = X^3 + 2X -2$. I want to find the splitting field of $f$ over $\mathbb{Q}$. My problem is that the roots of $f$ are too complicated, see Wolfram Alpha. How can I find this splitting field?
2
votes
0answers
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 ...
0
votes
2answers
46 views

Irreducibility of $8x^3 -6x +1$ in $\mathbb{Q}[x]$

I want to use Gauss' Theorem for that. So I have to show that $f(x) = 8x^3 -6x +1$ is irreducible in $\mathbb{Z}[x]$ which is equivalent to it not having any roots in $\mathbb{Z}$. Since any integral ...
3
votes
1answer
79 views

If $\min(\alpha,F)$ has only one root in $E$, must $\min(p(\alpha),F)$ have only one root in $E$

Let $F<E$ be an algebraic field extension. Let $\alpha\in E$ be such that $\min(\alpha,F)$ has only one root in $E$ (which will be $\alpha$).Is it true that for any $p(x)\in F[x]$ we must have: ...