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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5
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3answers
94 views

A good introductory book on Ring and Field theory with a view towards Number Theory ?

Please suggest some good introductory books on Rings&Fields with a view towards Number Theory ?
2
votes
1answer
50 views

What happens with $S_n$ in rings, integral domains and fields?

From Cayley's theorem we know that every group is a symmetric group, i.e. a group of permutations. But what happens when we "extend" a group to a ring or a field for example; is there any ...
1
vote
2answers
68 views

Showing a counterexample regarding normal extension

For field extensions K/E, E/F, if K/F is a normal extension, E/F is a normal extension also? I think this is false..but can't find a counterexample. Could anyone suggest me some example?
1
vote
0answers
11 views

Verification of proof that the left distributive property holds on a field of quotients, F

Here's my proof. I would like to know whether it is right or wrong and if the latter where it needs to be corrected: (a,b)=[(a,b)][(cf+ed,df)]=[(acf+aed,bdf)]. We want to show its equivalence to: ...
0
votes
0answers
30 views

the motivation of separable field extension

What is the origin of the motivation of separable field extension? Is it related to separable topological space or something else?
3
votes
3answers
91 views

If $F$ is an extension field of the field $K$ such that $[F:K] =1$, then $F=K$

Suppose $F$ is an extension field of the field $K$ such that $[F:K] =1$. How to prove that $F=K$? Thank you for your time and help.
1
vote
0answers
55 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
0
votes
1answer
18 views

Explaining a. Q(R,T) has unity even if R does not and b. In Q(R,T) every nonzero element of T is a unit

So I've been having trouble understanding this proof for quite a while now. I understand how the field of quotients is formed but not so much of why my professor's answers use this tactic for this ...
4
votes
0answers
45 views

Is there a recurrence formula for finding primitive polynomials in order to construct $\mathbb{F}_{p^n}$?

(In a previous question, I asked some related question but it was very disorientingly formulated. I apologize for that.) Let $p$ be a prime and $n\geq 1$ an integer. The finite field ...
0
votes
0answers
21 views

Embedding of splitting field for a family of polynomials

STATEMENT: Let $K$ be a splitting field for the family $\left\{f_i\right\}_{i\in I}$ and let $E$ be another splitting field. Any embedding of $E$ into $K^a$ inducing the identity on $k$ gives an ...
0
votes
1answer
28 views

Verification of proof that for a,b in ring R, assuming ab is a zero divisor at least one of a and b is zero divisor

I'm not so sure if this is correct but here's what I have so far: ab is a zero divisor iff there is a c$\neq$0 s.t. (ab)c=c(ab)=0 given ab$\neq$0 and c$\neq$0. Then we have ...
2
votes
0answers
51 views

Degree of the field extension for $x^4+3$

I want to find a splitting field for : $x^4+3 \in \mathbb{Q}[x]$ and the degree of that extension , I would be thankful if you verify/correct what I have tried for this problem : First I find the ...
1
vote
1answer
15 views

separable polynomial

How to show that if $K$ is a field of characteristic $p$ with $p$ prime and if $f(X)\in K[X]$ is an irreducible and inseparable polynomials, therefore there exist a $d\in\mathbb N, d>0$ such ...
2
votes
2answers
38 views

show that additive group of field of characteristic 0 is not cyclic

Show that additive group of field of characteristic 0 is not cyclic. If it is so then the additive group will be isomorphic to $\Bbb Z$ from here how do I proceed. I have seen Why must a field with ...
2
votes
1answer
43 views

More general constructible numbers?

I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is ...
1
vote
3answers
80 views

Calculate the dimension of the field extension $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}]$

I've though that $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}] = [\mathbb{Q}[ \sqrt2] : \mathbb{Q}].[ \mathbb{Q}[\sqrt2, \sqrt3]:\mathbb{Q}[ \sqrt2] ] $ And I know how to prove $[\mathbb{Q}[ \sqrt2] : ...
0
votes
0answers
35 views

Example of a local field of positive characteristic?

I am looking for a local field of positive characteristic, like $Q^{2}_{2}$ was used in this article: in fact, i need an another Example of a local field of positive characteristic like $Q^{2}_{2}$ ...
3
votes
1answer
49 views

$x^{p^2-1}-1$ is divisible by $x^8-1$ when $p$ is odd?

In the proof of reducibility of $x^4+1$ over $F_p$ (which is stated as a corollary of the structure theorem of the finite field $F_{p^n}$), the following implication is used in the Algebra by Dummit ...
1
vote
2answers
31 views

Which general methods of field construction do we know?

This question is partially motivated by: http://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory whereas I am not a specialist in field theory. Q1: Which methods do we ...
3
votes
1answer
73 views

Prove that any subfield of $\mathbb C$ must contain $\mathbb Q$

I just started reading Linear Algebra by Hoffman and Kunze, and I came across the following line: The interested reader should verify that any subfield of $\mathbb C$ must contain every ...
1
vote
1answer
28 views

What exactly is $k\left(T_{n}\right)_{n\in\mathbb{N}}$?

Let $k$ be a field and $T_{n}$ indeterminates over $k$. Is $k\left(T_{n}\right)_{n\in\mathbb{N}}$ the field of fractions of the form $x=\frac{p}{q}$, where $p\in k\left[T_{i}\right]_{i\in\mathbb{N}}$ ...
0
votes
1answer
24 views

Algebraically closed fields minimal

STATEMENT: This is a portion from Lang's proof of theorem 2.8 in chapter V section 2. If $E$ is algebraically closed, and $L$ is algebraic over $\sigma k$, then $\sigma E$ is algebraically closed and ...
0
votes
3answers
77 views

Why is $\alpha$ transcendental over $K$

Let $K$ be a field and let $\alpha$ be an element of the field $K(T)$ of rational functions, with $\alpha\not\in K$. Prove that $\alpha$ is transcendental over $K$. In this case $\alpha$ is of ...
1
vote
0answers
27 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
1
vote
3answers
42 views

Minimal Polynomial of $a +b\sqrt{2}$ as a function of a, b ∈ $\mathbb Q$

Determine the minimal polynomial over $\mathbb Q$ of $a +b\sqrt{2}$ as a function of a, b ∈ $\mathbb Q$. Let $x=a+b\sqrt{2}$ If $b=0$ then the minimal polynomial is $x-a$ if not, then ...
1
vote
1answer
34 views

Finite field extensions and minimal polynomial

I want to show the following statement: Let L/K be a finite field extension with $[L:K]=p$ for a prime $p$ Show that $[L:K]$ is simple Proof: 1) Choose $\alpha\in L$ with $\alpha \notin K$. Then ...
2
votes
1answer
70 views

Which one is a field?

Which one is a field? i) $\cfrac{\mathbb{Z}[x]}{\langle{x^2+2}\rangle}$ ii)$\cfrac{\mathbb{Q}[x]}{\langle{x^2-2}\rangle}$ I think both are correct because for both the cases ...
4
votes
5answers
153 views

Determine the minimal polynomial of $\sqrt 3+\sqrt 5$

I am struggling in finding the minimal polynomial of $\sqrt{3}+\sqrt{5}\in \mathbb C$ over $\mathbb Q$ Any ideas? I tried to consider its square but it did not helped..
0
votes
0answers
38 views

Degree of extension

Let $a, b \in \mathbb{C}$ and let $[\mathbb{Q}(a) : \mathbb{Q}]=m$, $[\mathbb{Q}(b) : \mathbb{Q}]=n$. Show that $[\mathbb{Q}(a,b):\mathbb{Q}]\leq mn$. If $(m,n)=1$ show that ...
0
votes
1answer
32 views

Splitting field of polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_7[x]$ and let $E \subseteq \overline{\mathbb{Z}}_7$ be the splitting field. Let $F \subseteq \overline{\mathbb{Z}}_7$ the splitting field of the ...
2
votes
1answer
42 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
0
votes
1answer
40 views

Show that $p=2^k+1$

When $p$ is an odd prime and $a=Re \left ( e^{\frac{2 \pi i}{p}} \right)$ then $[\mathbb{Q}(a) : \mathbb{Q}]=\frac{p-1}{2}$. Let $\theta = \frac{2 \pi}{p}$. If $\sin{\theta}$ is a constructable ...
1
vote
1answer
22 views

Show that the equation has exactly $m$ different roots in the algebraic closure


Let $n=p^rm$, where $p$ is a prime, $m \in \mathbb{N}, r \geq 0$ an integer and $(p,m)=1$. 
I have to show that the equation $x^n=1$ has exactly $m$ different roots in the algebraic closure ...
1
vote
1answer
37 views

Element from a formal power series that is algebraic over a field

I don't know how I should solve this exercise: The polynomial ring $K[T]$, and hence also its field of fractions $K(T)$, is a subring of $K((T))$. Give an example, for some field $K$ , of an ...
0
votes
1answer
29 views

Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$

Prove that: $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$ We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ...
0
votes
0answers
29 views

Symmetric group and fields

Let $K$ a field and $E=K(X_1,...,X_n)$ the fraction field of the domain of the polynomials $K[X_1,...,X_n]$. 1) Show that the symmetric group $S_n$ is a group of automorphism of $E$ 2) Show that ...
0
votes
1answer
11 views

Separable extensions-Need help

Let $K \leq M \leq E$ be field extensions, with $K \leq E$ separable. Show that the extensions $K \leq M$ and $M \leq E$ are separable. The extension $K\leq E$ is separable if all the elements in ...
-1
votes
2answers
50 views

If $\overline{f}(x)$ is irreducible in $\mathbb{Z}_p[x]$ then $f(x)$ is irreducible in $\mathbb{Z}[x]$

Let $f(x)=a_0+\dots +a_n x^n \in \mathbb{Z}[x]$. Let $p$ be a prime with $p \nmid a_n$. We define $\overline{f}(x)=\overline{a_0}+\dots +\overline{a_n} x^n \in \mathbb{Z}_p[x]$ How can I show that ...
0
votes
1answer
17 views

What is the relation between $Irr(a, F)$ and $Irr(a, K)$?

We have that $F \leq K \leq L$ and $a \in L$. If $a$ is algebraic over $F$ then it is also algebraic over $K$. What is the relation between $Irr(a, F)$ and $Irr(a, K)$? Let $Irr(a, K)=p(x) \in K[x]$ ...
0
votes
2answers
71 views

$\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$

I have to show that $\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$ and then I have to find $Irr(\sqrt{5}, \mathbb{Q})$. How can I show that $\sqrt{5} \in \mathbb{R}$ is algebraic over ...
0
votes
2answers
28 views

Why does it stand that #$\mathbb{Z}_p(a)=p^n$?

If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$ The proof is the following: Let $a \in K$. We take $\mathbb{Z}_p ...
2
votes
1answer
54 views

Real subfield of cyclotomic field is generated by $\zeta+\zeta^{-1}$

Let $p\neq 2$ a prime number, $\zeta=e^{\frac{2i\pi}{p}}$ and $\alpha=2\cos\left(\frac{2\pi}{p}\right)$. We consider the field extension $F=\mathbb Q(\zeta)$ and $E=F\cap \mathbb R$ of $\mathbb Q$. I ...
0
votes
1answer
22 views

How do we find the embeddings?

In my notes there is the following example: $$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id ...
2
votes
1answer
48 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
2
votes
1answer
34 views

Compute degree and Galois group of a Galois extension

I'm trying to show that $[L:\mathbb{Q}]=8$, where $L=\mathbb{Q}(i, \sqrt{2}, \sqrt{3})$. I tried using the tower law to show this by saying: $[L:\mathbb{Q}]=[L:\mathbb{Q(\sqrt{2}, ...
0
votes
1answer
30 views

How to find the splitting field?

How can I find the splitting field of $x^n+1 \in \mathbb{Q}[x]$ ?? If we have for example, $x^n-1 \in \mathbb{Q}[x]$ we would do the following: $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \dots +x+1)$$ So, the ...
1
vote
1answer
38 views

Field at which $f(x)$ splits [closed]

Let $f(x) \in \mathbb{Z}_p[x]$. Show that there is a finite field $\mathbb{F}_{p^n}$ at which $f(x)$ splits. And if $f(x)$ is also separable, show that $f(x) \mid x^{p^n}-x$. Could you give me some ...
0
votes
1answer
51 views

The union of finite field extensions is a finite field extension

Assume that all elements under discussion are algebraic over $F$. Let the notation "$K=F(A)$" mean that $A\subseteq K$ and there is an injective homomorphism $\sigma:F\to K$, and every element of $K$ ...
2
votes
4answers
137 views

Roots of different irreducible polynomials are algebraically independent

Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$. Let $\alpha$ be a root of some other monic irreducible $g\ne f$. Then is $f$ still irreducible in $F(\alpha)$? Is it true ...
1
vote
0answers
21 views

Field extensions and quotient fields

STATEMENT: Suppose that $F\subseteq E$ is a field extension of $F$. And assume $u\in E$ is transcendental over $E$. Then it readily follows that $F(u)\cong F(x)$, where $F(x)$ is the quotient field of ...