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
58 views

How to check field axioms given addition and multiplication tables

I need help with this question, i want to know the exact method of doing it with explanation. i am not able to get around with the logic of it.
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2answers
36 views

Write it as an element of this ring?

Since the degree of the irreducible polynomial $x^3+2x+2$ over $\mathbb{Q}[x]$ is odd, it has a real solution , let $a$. I am asked to express $\displaystyle{\frac{1}{1-a}}$ as an element of ...
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1answer
29 views

proof using field axioms only

I am studying the field axioms for real numbers from the book 'Tom m Apostol Calculus' and i am wondering if the following can be proved using the field axioms only. a=a and b=(b). the proofs that i ...
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1answer
36 views

Galois group of reducible polynomial

I want to find Gaolois group of $(x^3-x+1)(x^2+1)$ over $ \mathbb Q$. The polynomial of degree third is irreducible and has discriminant $-23$ so it's Galois group is $S_3$. Galois group of the other ...
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1answer
48 views

Algebraic element

$K \leq L, a \in L$ I am looking at the proof that if $a$ is algebraic over $K$, then $K(a)=K[a]$ : We show that $K[a]$ is a field, then we have that $K \subseteq K[a] \subseteq K(a) \subseteq L$. ...
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2answers
21 views

Homomorphisms between fields are injective.

How would I prove this? I know that I must show f(a)=f(b) => a = b I also know I must use the definition of homomorphism, ie: $f(a+b)=f(a)+f(b)$ $f(ab)=f(a)f(b)$ $f(1)=1$ I am assuming that a ...
0
votes
1answer
44 views

Proof that a given set is a field

I am solving the following exercise (linear algebra): show that: $\ \mathbb{Q}\lbrack\sqrt{2}\rbrack = \{ a + b\cdot\sqrt{2} \ \vert \ a,b \in \mathbb{Q}\} \subset \mathbb{R} \ $ is a field, with the ...
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1answer
46 views

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$.

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$. First, $\mathbb Q(z)\subseteq \mathbb Q(z^3,z^5)$ would be trivial, right? Then we ...
3
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1answer
57 views

Compute the transcendence degree (transcendence degree and tensor products)

$\DeclareMathOperator{\quot}{Quot}\DeclareMathOperator{\tr}{tr}$ Let $I_1$ and $I_2$ be nontrivial ideals in $\mathbb C[x_1,\ldots,x_k]$ and $\mathbb C[y_1,\ldots,y_m]$, respectively. Define $$ R_1 ...
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2answers
29 views

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$.

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$. I understand that $\mathbb Q(\sqrt{a})$ is the smallest ...
4
votes
2answers
76 views

For what natural numbers $n$ is $\mathbf Z/n\mathbf Z$ $[x]/(x^3+x+1)$ a field?

I recently saw this question in the exam of a first abstract algebra course in my college. It shouldn't be too difficult, yet I can't seem to get the solution. Any ideas on how to tackle this?
1
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2answers
67 views

How to show rational function field of an affine subvariety with dim>0 is not algebraically closed?

I do not know how to show the following statement. If $X\subset A^n$ is an irreducible subvariety, $\dim X>0$, then the rational function field of $X$, $K(X)$ is not algebraic closed. What ...
0
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2answers
59 views

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$.

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$. One thing to note is $a^{-1}\ne \large\frac{1}{a}$ (same goes for $b$) in this instance as there could be fields where this isn't ...
5
votes
1answer
105 views

History of the three “impossible” compass-and-straightedge problems

I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic. I wanted to know if the classical Greek problems (doubling the cube, ...
25
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1answer
531 views

Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true: Let $x,y$ be real, irrational ...
3
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1answer
39 views

Can a $p$-adic field admit a different valuation?

Let $L/\mathbb{Q}_p$ be a finite extension. Question: Is it possible for $L$ to admit a henselian valuation with residue characteristic $q \not=p$? I would think surely not, but I can't see a ...
1
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1answer
32 views

Isomorphisms between $\Bbb R^n$ and fields

Are there real vector spaces with dimension $\geq 3$ that are isomorphic to a field? I case $n=2$ there are the complex numbers and for $n=3$ the quaternions are non-commutative. Thanks in advance.
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1answer
95 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.
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2answers
33 views

Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
3
votes
2answers
70 views

In the theorem is it necessary for ring $R$ to be commutative?

According to the statement of theorem that a commutative ring $R$ with prime characteristic $p$ satisfies $$\begin{align} (a+b)^{p^n} = a^{p^n} + b^{p^n} \end{align}$$ $$\begin{align} (a-b)^{p^n} = ...
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0answers
20 views

Is every field between $F$ and $F(\alpha_1,\cdots,\alpha_n)$ of the form $F(\alpha_j,\cdots,\alpha_k)$

Say I have a field $F$, and an extension field $L = F(\alpha_1,\cdots,\alpha_n)$. Is it true that every $K$ such that $$ F \subset K \subset F $$ (all field extensions), $K = ...
1
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1answer
31 views

$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
0
votes
2answers
33 views

Algebraic Extensions

I have the following question: there is this statement i can't understand: Let $A$ be an integral domain which is integrally closed ( in its field of franctions ) and let $K$ be its fraction field. ...
0
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1answer
31 views

If $K$ finite field of order $p^8$ where $p\ne3$ then $\sum_{\alpha \in K}{\alpha^2} = 0$

Let $K$ be finite field of order $p^8$ where $p\ne3$ is a prime. Show that $\sum_{\alpha \in K}{\alpha^2} = 0$.
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3answers
23 views

Isomorphism of field extension

Let $F$ be a field. I need to prove that if $\sigma$ is an isomorphism of $F(\alpha_1,...,\alpha_n)$ with itself such that $\sigma|_F = id_F$ and $\sigma(\alpha_i)=\alpha_i$ for $i=1,...n$, then ...
0
votes
1answer
36 views

(Updated): Showing that a set $M$ with two elements classifies as a field

My question is more conceptual, so I will come straight to the exercise: Exercise: Let $M= \lbrace g,u \rbrace $ be a Set. On $M$ the Addition and the Multiplication is given by: \begin{align} ...
6
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0answers
22 views

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, What is the intersection $F_\infty\cap K_\infty$? Here $\zeta_{2^n}$ is a ...
4
votes
2answers
63 views

If $f(x)\in\mathbb{Q}[x]$ of degree $p$ and $\operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$ then $f(x)$ is irreducible.

Let $f(x)\in\mathbb{Q}[x]$ , $p$ prime, $\deg f(x)=p$ and $G = \operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$, where $K$ the the splitting field of $f(x)$ over $\mathbb{Q}$. Show that ...
0
votes
1answer
45 views

Question about $\gcd$

Theorem: Let $K$ be an infinite field and let $L:=K(\alpha, \beta)/K$ be a field extension with $\alpha$ algebraic over $K$ and $\beta$ separable over $K$. Then $L = K(z)$ for a certain $z \in L$. ...
1
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1answer
39 views

What is the difference between the algebraic function fields and the fields itself

I'm studying this book and I don't understand exactly what's the difference between the algebraic function field $F/K$ and $F$ itself. Thanks
0
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1answer
74 views

F be the smallest subfield of the real numbers which contains irrational a. Prove that F is countable.

Let a be an irrational number and let F be the smallest subfield of the real numbers which contains a. Prove that F is countable.
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1answer
38 views

How to Prove: If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.

I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go ...
0
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1answer
32 views

On an example of an inseparable element in a field extension

Put $F := \mathbb{F}_p(t) = \left\{\frac{f(t)}{g(t)} : f(t), g(t) \in \mathbb{F}_p[t], g(t) \neq 0\right\}$. Now in one book the author considers the so-called Frobenius homomorphism $\mathcal{F}:F ...
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4answers
44 views

Easy question about tower of fields

If $E/F/G$ is a tower of fields and $[E:G]<\infty$, then does $[F:G]<\infty$? I suspect the answer to be "yes", but somehow the fact that a basis of $E$ over $G$ might have vectors in ...
1
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0answers
36 views

Denseness of algebraic and transcendental elements in R [NBHM 2014]

Which of the following statements are true? a. Algebraic numbers over Z are dense in R. b. Transcendental numbers over Z are dense in R. Let me write what I did. For a), the concerned set is a subset ...
0
votes
2answers
36 views

Intersection of any family of subfields is itself a subfield

Prove that the intersection of any family of subfields is itself a subfield. In the countable case: Suppose that $\mathscr K$ is a field and consider $(\mathcal K_n)_{n\in\mathbb N}\subset \mathscr ...
1
vote
1answer
39 views

$\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$.

I need to show that $\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$. My approach was to find a bigger proper ideal containing $f(x,y)$ but i am unable to ...
3
votes
1answer
86 views

Elements of $GL_{2}(\mathbb{Z})$ of finite order

Prove that any element of $GL_{2}(\mathbb{Z})$ of finite order has order $1,2,3,4,6$ using FIELD THEORY. My idea is to reduce such a finite order matrix say $A$ with order $n$ to modulo a prime $p$. ...
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0answers
14 views

Trascendental extension [duplicate]

Let $p$ a prime number and let $\mathbf{F}_p(X)$ the field of rational functions over $\mathbf{F}_p$. The degree $[\mathbf{F}_p(X):\mathbf{F}_p(X^p)]$ is $p$? I'm a little confused; thanks.
2
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1answer
18 views

Square root of one of $2,3,6$ exist in a prime field

Using Gauss's law of Quadratic Resiprocity it is immediate that one of $2,3,6$ is a square in $\mathbb{F}_{p}$. I am looking for a solution which uses basic field theory only. I was thinking of ...
1
vote
3answers
181 views

Can we say “commutative ring = field”?

We know the difference between ring (R) and field (F) is that R does not guarantee multiplication is commutative. Now, if considering commutative R, which means (R,.) is a group, can we say: ...
0
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1answer
20 views

Let K/F be a transcendental extension , then does every F-homomorphism has to be an automorphism?

I have figured out that if $K/F$ be an algebraic extension , then does every F-homomorphism need not be an automorphism . But I can't figure it out for in thee trasncendental case.
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0answers
41 views

Find $[\mathbb{Q}(\sqrt [3] {3},\sqrt [3] {2}):\mathbb{Q}]$ [duplicate]

I am trying to find $[\mathbb{Q}(\sqrt [3] {3},\sqrt [3] {2}):\mathbb{Q}]$. My guess is it is $9$. There are 3 possibilities-3,6,9. If it is not 9 then $X^{3}-3$ is not irreducible over ...
3
votes
3answers
62 views

$[\mathbb{Q}(\sqrt [3] {2}+\sqrt {5}):\mathbb{Q}]$ [duplicate]

What is $[\mathbb{Q}(\sqrt [3] {2}+\sqrt {5}):\mathbb{Q}]$? A straight forward way will be to just set $x=\sqrt [3] {2}+\sqrt {5}$, take powers and reach at a polynomial in $x$ and show the polynomial ...
5
votes
1answer
89 views

$\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?

Is there a general or elegant way to approach this problem? One can show that $\sqrt 5+\sqrt[3] 2$ is a root of the hexic $x^6-6x^4-10x^3+12x^2-60x+17$, which should then be its minimal polynomial to ...
0
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0answers
29 views

Field isomorphism and order of elements

I know that group isomorphism preserves order of element but can someone plese tell me does field isomorphism preserves order of elements?
2
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0answers
124 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
0
votes
2answers
48 views

every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
0
votes
1answer
31 views

Prove $\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$

I want to know why the following two are equivalent: $$\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$$, where $\mathbb{Q}$ is the rational number field, and ...
1
vote
2answers
38 views

Is $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ a splitting field of some polynomial

Is it true that the extension $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ is the splitting field of some polynomial over $\mathbb{Q}$? My guess is no. But I can not prove it. Some observations I made are as ...