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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Question in Algebraic closed field

I'd like to know how to prove algebraic numbers form a field, i.e, if $a,b$ are 2 algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...
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3answers
51 views

Question about field extentions?

if $\mathbb{Q}(\sqrt{3}) $ can be looked at as the field of rational numbers with $\sqrt{3}$ appended to it, and can be furthermore looked at like $\mathbb{Q}[x]/x^2 - 3$ what does a field extention ...
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2answers
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Finding the conjugates, why can they argue this way?(exercise)

In one exercise I am supposed to find the conjugate of $\sqrt{2}+i$ over $\mathbb{Q}$. I found the answer by finding irr$( \sqrt{2}+i,\mathbb{Q})$, and then solving the polynomial finding all the ...
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6answers
66 views

Prove this polynomial falls within $\mathbb R[x]$

[ The problem below is from Yao Musheng (姚慕生), Wu Quanshui (吴泉水), Advanced Algebra (高等代数学) Ed $2$, Fudan University Press, page $207$. ] Suppose $f(x)\in \mathbb C[x]$. If $\forall c\in \mathbb ...
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3answers
58 views

Show that $\sqrt{5} \notin \mathbb{Q}(\sqrt{3})$

Show that $\sqrt{5} \notin \mathbb{Q}(\sqrt{3})$. I don't even know where to start. I can't find references to this in my textbook anywhere. I feel like the notation came out of nowhere.
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2answers
533 views

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$? If $\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in\mathbf{Q}\}$ and $\mathbf{Q}(\sqrt{6})= \{a+b\sqrt{6} | ...
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1answer
54 views

Why can't Eisenstein Criterion be used for certain polynomials (to show that it's irreducible over $\mathbb{Q}$)?

Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$? I mean even if we were to apply Eisenstein here, there doesn't exist a prime ...
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3answers
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2
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1answer
28 views

What is the automorphism group of the field $\mathbb{Z} /p\mathbb{Z}(t)$?

Here, $t$ is transcendental over $\mathbb{Z} /p\mathbb{Z}$. How big is this group? What are its elements? Is for example the map $t \to -t$ an automorphism?
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2answers
43 views

Field that is a subfield of own of its subfields

Let $K$ and $L$ be fields. We have homomorphisms $f: K \to L$ and $g: L \to K$. Are $K$ and $L$ necessarily isomorphic?
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1answer
43 views

Automorphisms (in the context of Galois Theory)

Can someone give an explanation of what an automorphism is, in the context of Galois Theory? I keep thinking it is the set of maps which send roots of polynomials to their conjugates but I feel that ...
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1answer
37 views

Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
2
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1answer
46 views

Does all splitting fields have characteristic 0?

Does all splitting field have characteristic 0? This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with ...
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2answers
18 views

Surd-like trinomials form a field

This is a problem from Artin's book "Algebra". In the fifth miscellaneous problem of the chapter "Vector spaces", he has asked to prove that: If $\alpha$ is a cube root of $2$, then the real numbers ...
3
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1answer
30 views

Algebraic or not algebraic extension?

Suppose $F^{\prime}/F$ is an algebraic extension of fields, and that $F^{\prime}$ is a finite field extension of $K^{\prime}(x^{\prime})$, where $x^{\prime}$ is transcendental over $K^{\prime}$, and ...
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0answers
34 views

Can it be proved that this extension is algebraic?

Assume that we have a field F, an extension field E of F, and both of them are contained in the algebraic clousure $\overline{F}$. Let E have the property that every automorphism of $\overline{F}$ ...
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0answers
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Dimension of compositum of two fields, one of them Galois.

Let $L, K, F = L \cap K$ be fields such that $[L:F] , [K:F] < \infty$. $LK$ is defined as the compositum of the two fields(shortest field containing $K$ and $L$ in some algebraic closure). Also, ...
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1answer
21 views

Extension E/K such that E/F is a splitting field

The question asks us to prove that there is an extension $E/K$ such that $E/F$ is a splitting field of some polynomial $f(x) \in F[x]$ where $K/F$ is a finite extension. I'm not really sure how to ...
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1answer
58 views

If $d \equiv 1 \pmod 4$, is $\mathbb Q[\sqrt d]$ the field of fractions of $\mathbb Z\left[\frac{1+\sqrt d}{2}\right]$?

If $d \equiv 1 \pmod 4$, is $\mathbb Q[\sqrt d]$ the field of fractions of $\mathbb Z\left[\frac{1+\sqrt d}{2}\right]$? Is $\mathbb Q\left[\frac{1+\sqrt d}{2}\right]$? I am confused about quadratic ...
1
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1answer
26 views

Primitive element theorem, simple extension

Let $X$, $Y$ be indeterminates over $F_2$, the finite field with 2 elements. Let $L = F_2(X, Y )$ and $K = F_2(u, v)$, where $u = X + X^2$, $v = Y + Y^2$. Explain why $L$ is a simple ...
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1answer
22 views

Proof help: why is the constructed field a splitting field?

Here is my books definition of a splitting field: Note that it uses the word: smallest: In the last converse part of this theorem. I see that the field E created is a field that contains F(this is ...
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5answers
85 views

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I mean, $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ can be viewed as: $\mathbb{Q}[\sqrt{2}][\sqrt{3}]$, as polynomials of ...
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3answers
44 views

Showing that $\mathbb{Z}_N$ is a field if $N$ is prime

I know that $N$ being prime is a necessary and sufficient condition for $\mathbb{Z}_N$ to be a field. I know how to prove that it's necessary but I'm not sure how to prove that this is a sufficient ...
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0answers
43 views

Galois extension over power series fields

Let $K$ be a field, and $L$ be an algebraic extension of $K$. I think it is known that if $T$ is a finite extension of $K((X))$, then $T$ is complete with respect to the $X$-adic valuation, hence if ...
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1answer
26 views

Showing that $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ irreducible over $\mathbb{Q}(\sqrt[4]{5})$

I am unsure how to show that the polynomial $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ is irreducible over $\mathbb{Q}(\sqrt[4]{5})$. If it were reducible, it would have a root in ...
5
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0answers
68 views

algebraically closed fields of characteristic 0 and $\mathbb{C}$

Let $k$ be an algebraically closed field of characteristic 0. Then I've heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding ...
8
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0answers
66 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
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2answers
26 views

If $ f \in \mathbb{Q}[x] $ is irreducible and has a root in $ \mathbb{Q}(\sqrt{2}, i) $, then it splits

I'm trying to find a solution for the following problem: let $ f \in \mathbb{Q}[x] $ be irreducible. Suppose $ f $ has a root in $ \mathbb{Q}(\sqrt{2}, i) $. Prove that $ \deg f \in \{1,2,4\} $ and ...
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2answers
39 views

Degree of a splitting field over $ \mathbb{Q}$

I'm trying to solve the following problem: Let \begin{equation*} f(x) = x^4 - 2x^2 - 2 \in \mathbb{Q}[x] \end{equation*} and $ E $ be its splitting field. What is the degree $ [E: \mathbb{Q}] $? ...
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0answers
21 views

Action of the group ring $\mathbb{Z}[\text{Gal}(K/\mathbb{Q})]$ on the field $K$

Let $K$ be an algebraic number field, let $G$=Gal($K/\mathbb{Q}$). Let $\mathbb{Z}[G]$ be the group ring, or the set of formal sums $$\left\lbrace\sum a_i\sigma_i : a_i\in \mathbb{Z}, \sigma_i \in ...
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0answers
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Notion of Separability

Notion of Separability An irreducible polynomial $f\in F[X]$ is separable, if $f$ has no repeated root in a splitting field, if $f$ is not necessarily irreducible, then we call $f$ separable, if ...
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2answers
521 views

Finding the minimal polynomial in this field extension of $\mathbb Q$?

I have a field extension \begin{equation*} K = \mathbb Q[x]/(x^2 - 5) \end{equation*} of $\mathbb Q$, and an element $a = \bar x \in K$. I need to find the minimal polynomial of $a$ over $\mathbb ...
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2answers
120 views

Is the intersection of finite index subfields finite?

Suppose that $K$ and $L$ are two fields contained in some larger field, and let $KL$ denote the smallest subfield of the ambient field containing both of them. If $KL$ is a finite extension of both ...
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0answers
19 views

unramified extension of valued fields

I came across the following exercise: Let $M$ be a valued field with subfields $E$ and $L$, and suppose that $L$ is finite over some field $K\subseteq L\cap E$. Show that $EL/E$ is unramified if ...
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2answers
50 views

Why is $\mathbb{Q}$ left fixed?

In this example I get that 1 is left fixed, because every multiplicative element of an isomorphism is left fixed?, but why is $\mathbb{Q}$ left fixed as a consequence of this?
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6answers
192 views

Does $\mathbb Q(\sqrt{-2})$ contain a square root of $-1$?

This isn't a homework question but one I found online. Does $\mathbb Q(\sqrt{-2})$ contain a square root of $-1$? We just started doing field theory in my class and I want extra practice, but I ...
2
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1answer
67 views

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$ where $p$ is a prime number. My thoughts are: I am lost My intuition says it has to be $ \frac{p-1}{2}$ and ...
2
votes
1answer
35 views

Show for any prime $p$ and $a \in \mathbb{F}_{p}$ that $x^p-a$ has multiple roots

Show for any prime $p$ and $a \in \mathbb{F}_{p}$ that $x^p-a$ has multiple roots using the derivative of $x^p-a$ which is $px^{p-1}$ if they are relatively prime then $x^p-a$ only has simple roots. ...
0
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1answer
36 views

Is a field just a commutative ring? [duplicate]

Is a field just a commutative ring? My algebra professor didn't give a very wide introduction to this algebraic structure, and I did not get a real grasp of what a field is. We're studying ...
2
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1answer
37 views

Multiple roots in $\mathbb{Z}_p$

Let f(x) ∈ $\mathbb{Z}$[x], a polynomial of degree n. Suppose f(x) has n distinct roots $a_1, ..., a_n$ ∈ $\mathbb{C}$. Now, with a given f(x), we call a prime p "bad" if f(x) has a ...
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0answers
28 views

Extension Field of $\mathbb{Q}$ and its Galois group

How many elements are in $Gal(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}:\mathbb{Q})$?
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0answers
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On transcendence base and separability

This is a problem in Hungerford's Algebra. Let $k$ be a perfect field and $F$ an extension field of $k$ with transcendence degree 1 and $F$ is not perfect. We have to show that $F$ is separably ...
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0answers
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About a field extension and its normal closure

Is the extension $K=\mathbb Q(\root5\of2)$ over the rationals normal? If not find its normal closure. I know that K is not normal but I can not show and I think normal closure of $K$ is ...
2
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0answers
28 views

Subgroups of Galois group and intermediate fields lattice for $(x^3-2)(x^2-3)$

I am trying to systematically determine all subgroups of Galois group and intermediate fields for $(x^3-2)(x^2-3)$(over $\mathbb Q$). It's not hard to determine the Galois group of $(x^3-2)$ and ...
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1answer
30 views

Finite fields and cardinality

I am trying to get my head around the proof of the following: Suppose K is a finite field. With $p=charK, |K|=p^r$ where r is a positive integer. I am supplied with the following proof: I do not ...
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3answers
47 views

Find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$

Consider $a\in\mathbb{R}$ and $x^3-x+a=0$ with $x_{1,2,3}\in\mathbb{C}$. We need to find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$. It seems be equivalent with to find a such that ...
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0answers
24 views

Linear recursions in finite fields

Let $F$ be a finite field and let $\alpha$, $\beta$ be distinct nonzero elements of $F$. Let $\alpha$ have order $r$ and let $\beta$ have order $s$. Let $M = \operatorname{lcm}(r, s)$. Let $a,b$ be ...
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2answers
48 views

field of fractions and being algebraically closed

prove that for every field $F$ the field of fractions $F(x)$ is not algebraically closed. it is a problem which i don't know how to deal with it. help please. thank you.
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1answer
18 views

separable polynomial $ \bmod p$ (definition)

Given a polynomial $ f(x) \in K[x]$, where $K$ is a number field, we say that $f$ is separable if all its roots are distinct in an algebraic closure of $K$. Question: What does it mean a ...
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1answer
17 views

Polynomial with degree less than degree of an irreducible polynomial of the same root is 0

Let $F$ be a field, and $p(x)\in F[x]$ be an irreducible polynomial. Suppose $\alpha$ is a root of $p(x)$. Show that if $q(x)$ is a polynomial such that $\deg q(x) < \deg p(x)$ and $q(\alpha)=0$, ...