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
31 views

If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Let $a$ and $b$ be elements in extension field $F$. Is it true that: If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$? I just did the same ...
2
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1answer
26 views

Is the formula $[k(a,b):k][k(a) \cap k(b):k] = [k(a):k][k(b):k]$ true for simple algebraic extensions ($k(a)/k$ and $k(b)/k$ )?

Let $k$ be any field and $L/k$ be a field extension. Suppose $a, b \in L$ are algebraic over $k$. Is the formula $[k(a,b):k][k(a) \cap k(b):k] = [k(a):k][k(b):k]$ true? This formula comes from page ...
2
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1answer
39 views

Finite separable field extensions such that $KL/L$ and $K/K\cap L$ have non-isomorphic automorphism groups

If $K/F$, $L/F$ are finite separable extensions (not necessarily finite Galois extensions), then it seems clear that $KL/F$ is also a finite separable extension. However, in this case, is ...
4
votes
1answer
72 views

Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?

Consider the function field $k(x,\sqrt{1-x^2})$ of the circle over an algebraically closed field $k$. Is $k(x,\sqrt{1-x^2})$ a purely transcendental extension over $k$? I'm curious because I was ...
1
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1answer
81 views

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...
3
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1answer
22 views

Find the number of integers $r$ such that the polynomial $x^{r}-a$ has a linear factor over $\mathbb{F}_{p^{n}}$

If we have a finite field $\mathbb{F}_{p^{n}}$, how does one determine the number of integers $r$ in $\{0,1, \ldots, p^{n}-2 \}$ for which the equation: $x^{r}=a$ has a solution for every $a \in ...
0
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1answer
12 views

Prime subfield equivalent definitions

So I have field $F$ (any characteristic), and its prime subfield $K$. I have three definitions: (i) that $K$ is the subfield of $F$ such that $K$ has no proper subfield; (ii) that $K=\bigcap_i K_i$ ...
2
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1answer
53 views

Which Galois Field is isomorphic to this extension?

Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$? [Adding the following ...
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1answer
23 views

Reducing a sum of products of primitive nth roots

QUESTION: Suppose $\zeta$ is an primitive n-th root of unity. And suppose $n=p^r$ where $r\geq 1$. What is $(1-\zeta)\zeta^{p^r-p^{r-1}-1}$ written as the sum of the basis elements ...
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3answers
89 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
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1answer
46 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
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2answers
66 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?
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0answers
9 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
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0answers
27 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
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3answers
89 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.
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0answers
48 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
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1answer
13 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
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0answers
41 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
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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
26 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 ...
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0answers
35 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
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1answer
14 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
35 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
35 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
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3answers
73 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
29 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
47 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 ...
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2answers
29 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
65 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
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1answer
27 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
22 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
67 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
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0answers
25 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
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3answers
38 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
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1answer
26 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
68 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
135 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
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0answers
29 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
25 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
36 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
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1answer
37 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
19 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
35 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
28 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 ...
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0answers
28 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
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1answer
8 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 ...
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votes
2answers
47 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
15 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
68 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
24 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 ...