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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2answers
411 views

minimal field extension of Q($\sqrt[3] {2}$)

I need to describe the minimal field extension $\mathbb Q(\sqrt[3] {2})$ of the rational numbers $\mathbb Q$ that contain $\sqrt[3] {2}$. $\mathbb Q(\sqrt[3] {2}) =\{a+b\sqrt[3] {2}+c(\sqrt[3] ...
3
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3answers
1k views

Prove that Q($\sqrt{2}$, $\sqrt{3}$) is a field

Prove that $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \{a+b\sqrt{2} +c\sqrt{3} +d\sqrt{6}\ |\ a,b,c,d \in \mathbb{Q}\}$ is a field. I am doing the subfield test, but having trouble in showing how to express ...
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2answers
910 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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0answers
78 views

Galois extension and subfield

Let $E/F$ be a Galois extension and $Gal(E/F)\cong \Bbb Z/p^3\Bbb Z$. Assume that there exists subfield $K$ of degree $p$ of $E$. (i.e, $[E:K]=p$) Then, show that any proper subfield of $E$ is ...
1
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1answer
259 views

Show C(X) is a vector space over $\mathbb R$ with the following operations?

I have a set of continuous functions, $C(X): X \rightarrow R$ on a compact metric space, and definitions of addition & multiplication: $$(f+g)(x) = f(x)+g(x)$$ $$(\lambda f)(x) = \lambda ...
6
votes
3answers
678 views

Brauer group of a field of rational numbers

Can we say anything about Brauer group of $\mathbb{Q}$? And how can we construct it?
4
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0answers
116 views

Multiplicative nature of the separability degree

In what follows, let $E / F$ be an algebraic extension, $h(x),f(x)\in F[x]$ polynomials, $h(x)$ irreducible. Definitions. We say $h(x)$ is separable if it have not repeated factors. We say $f(x)$ ...
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2answers
64 views

If two elements a and b in a field extension L/K are algebraic in K and have the same minimal polynomial, prove that $K(a)=K(b)$

I have a field extension $L/K$ and two elements $a,b \in K$, which are algebraic over $K$ and both have the same minimal polynomial, and need to prove that then $K(a)=K(b)$. I can see how this can be ...
3
votes
2answers
365 views

Prove that $\mathbb Q(\sqrt 2) \neq \mathbb Q(\sqrt 3)$

I've tried writing out the contents of each and attempting to get a contradiction by equating arbitrary elements but can't get this to work. I can't think of any counterexamples as everything I come ...
5
votes
5answers
1k views

Prove that $\mathbb Q(\sqrt 2, i) = \mathbb Q(\sqrt 2 + i)$

I probably don't understand the definition of each properly. What I am thinking is that $\mathbb Q(\sqrt 2, i)$ has elements of form $a_1(\sqrt2) ^1 + a_2(\sqrt2) ^2 + ... + a_n(\sqrt2) ^n + a_1i^1 ...
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1answer
162 views

R ring is noetherian, commutative, unitary and integral domain, is R a field?

This is the question: "let R be a commutative unitary ring that is also integral domain and noetherian, prove that R is a field" I'm having some trouble proving this. For R to be noehterian means ...
1
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0answers
62 views

Determing the structure of the subgroup of an automorphism group

Suppose we have two automorphisms of an extension field $L=\mathbb{Q}(t)$ for some variable, given by $\sigma: t \mapsto 1-t$ and $\tau : t \mapsto \frac{1}{t}$. Clearly $\langle \sigma , \tau ...
1
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1answer
53 views

What is the underlying math in this relation?

Suppose we have the constraint $$.7x_1+.4x_2+.5x_3<1,$$ $$x_1,x_2,x_3\in\{0,1\}$$ Then we can convert it to a Boolean expression with binary variables of the form $$(\neg x_{1}\wedge\neg ...
3
votes
2answers
241 views

Tensor product of a field with itself.

I am proving the fact that if $A$ and $B$ are two central $k$-algebras where $k$ is a field (so then $Z(A) = Z(B) = k$), then $A \otimes B$ is also central. I made almost everything except this: I ...
8
votes
5answers
353 views

Does there exist a field $K$ such that $\mathbb R \subsetneq K \subsetneq \mathbb C$?

I'm thinking of unions of $\mathbb R$ with some subset of $\mathbb C$ but am not sure how to approach this without ending up with all of $\mathbb C$. Doe anyone have any suggestions?
2
votes
1answer
122 views

Norm of an integral element is integral

Let $L / K$ be a finite field extension and let $x \in L$. Let $Norm_{L/K}(x)$ be the determinant of multiplication by x over L. Now, let us assume that $A \subset B$ are rings such that $K = Frac(A)$ ...
1
vote
1answer
96 views

Let $K/\mathbb{F}_2$ be the splitting field of the polynomial $x^{17}-1$. Determine $[K:\mathbb{F}_2]$

Mainly, I would like to check if I am correct. Over $\mathbb{Q}$, the result is shown in Sec. 13.4 or 13.6 of Dummit and Foote. I think the result is essentially the same here. Since $x-1$ is a ...
1
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1answer
63 views

Convert a Boolean expression to a linear expression?

Suppose we have a Boolean expression $$(\neg x_{1}\wedge\neg x_{2})\vee\left(\neg x_{1}\wedge\neg x_{3}\right),$$ which we need to be true. Is there a method to convert this to a linear expression of ...
3
votes
1answer
120 views

Polynomials having as roots the sum (respectively, the product) of two algebraic elements

This question remained somehow incompletely solved. The OP also asked for an explicit form of the minimal polynomial of the sum (respectively, product) of two algebraic elements (in a field ...
4
votes
0answers
686 views

The automorphism group of a field with $p^2$ elements

Suppose $K$ is a field. Then we call $f: K\to K$ a (field) automorphism if $f$ is a one-to-one, onto and unital (i.e. $f(1)=1$) homomorphism of rings. The following results are well-known. There ...
5
votes
0answers
124 views

Irreducible polynomials of degree $d$ such that $[x]$ generates $\mathbf{F}_{p^d}^{\times}$

It is often convenient to represent the field $\mathbf{F}_{p^d}$ as $\mathbf{F}_p[x]/(f(x))$, where $f$ is irreducible with degree $d$, $f$ has just a few nonzero terms, and $[x]$ itself is the ...
0
votes
1answer
401 views

Conjugate isomorphism of the automorphism groups of two field extensions?

This is the problem I'm trying to solve: Let $K/F$ be a field extension, and let $f: K → K'$ be an isomorphism of $K$ with a field $K'$ which maps $F$ onto the subfield $F'$ of $K'$. Prove that ...
1
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2answers
73 views

Is it possible for two finite fields with the same characteristic to not be isomorphic? [duplicate]

Examples of this? I know this is possible with infinite fields but I'm not sure about finite fields.
3
votes
1answer
134 views

Find the order of a polynomial

I want to find the cycle set for the polynomial $p(x)=x^{23}+x^6+1$ over $\mathbb{F}_2$. So, I have the connection polynomial $C(D)=1+D^{17}+D^{23}$ over $\mathbb{F}_2$ The factors to $C(D)$ are: ...
3
votes
1answer
191 views

polynomials factorization over rings and finite fields

Any nonzero polynomial over a subring $R$ of $\mathbb{C}$ is a product of irreducible polynomials over $R$. And for any subfield $K$ of $\mathbb{C}$, factorization of polynomials over $K$ into ...
6
votes
2answers
578 views

Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.
5
votes
1answer
124 views

bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only?

I was wondering, can you define a bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only ($a \in \mathbb{Q}$)? Of course there are many set theoretic bijections like ...
1
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1answer
142 views

Find solutions to equation (ring/field theory, residue class)

I'm trying to solve this problem: A residue class ring mod $n$ is a field if n is prime. Let $\mathbb{Z}_p$ be a residue ring, p prime. Let $a \in \mathbb{Z}_p$. What are all solutions $x \in ...
2
votes
1answer
30 views

When can complete dense linear orders be made into topological fields?

By a "complete dense linear order," I mean a dense linear order in which every nonempty subset with an upper bound has a least upper bound. The canonical example, $\mathbb{R}$, is a topological field ...
1
vote
0answers
105 views

Independent indeterminate roots and coefficients of a polynomial

Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of ...
1
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1answer
87 views

Addition in finite fields

For a question, I must write an explicit multiplication and addition chart for a finite field of order 8. I understand that I construct the field by taking an irreducible polynomial in $F_2[x]$ of ...
0
votes
0answers
43 views

$p > 2$ and ramification of archimedean places

Fix a rational prime $p$. I know that for a $p$-extension (ie. a Galois extension of degree a power of $p$) of an algebraic number field $k$, some places can not ramify: complex places cannot ...
4
votes
1answer
134 views

Finite extensions of fields that are algebraically closed

Consider a finite field extension $K/k$ with $K$ algebraically closed. The immediate example that comes to mind is $\mathbb{C}/\mathbb{R}$, of degree $2$. Are there any other examples? Can we ...
0
votes
1answer
143 views

Determine the degree of another field extension.

I am looking for more details to part of a solution to this question How do I find a splitting field $x^8-3$ over $\mathbb{Q}$?. I would like to determine the degree of the splitting field for ...
0
votes
0answers
59 views

Number rings and (round) parentheses versus (square) brackets [duplicate]

Is there a reason for the difference in the use of parentheses versus brackets as used in algebraic extensions. For example, when the field rational numbers ${\mathbb{Q}}$ extended with $i = ...
4
votes
2answers
49 views

A basic theorem on field isomorphisms

I'm reading A Book Of Abstract Algebra by Charles C. Pinter. On page 314 is the following theorem: Let $h:F_1\to F_2$ be an isomorphism, and let $p(x)$ be irredicible in $F_1[x]$. Suppose $a$ is a ...
2
votes
1answer
38 views

Equations over fields

Let $x_1,\cdots, x_n$ be distinct elements of a given field $F$ such that for any $k$, $\sum_{i=1}^n x_{i}^k = 0$. I want to show that all $x_i$'s are zero.
2
votes
3answers
452 views

Ideals in a real/complex number field?

Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$. Quick web search gave no ...
0
votes
1answer
45 views

Kronecker's Theorem - what can be deduced if $f$ is reducible?

I am happy with the statement of Kroncker's Theorem as follows. Let $\:$ $ f \in K[X]$. Then there exists a simple field extension $L = K(\alpha)$ of $K$ with $f(\alpha) = 0$. If $f$ is irreducible ...
3
votes
1answer
92 views

Radical extension over $\mathbb{Q}$

Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d$. Prove that $E=\mathbb{Q}(\sqrt[d]{a})$. ...
2
votes
2answers
164 views

How to show field extension equality

I've seen similar field extension questions on SE, but nothing with a third root, and I'm having trouble adapting any of those solutions to this problem. So I'm trying to prove that ...
2
votes
1answer
122 views

Mathematics and Origami

I am reading through this paper about the math behind origami: http://www.math.washington.edu/~morrow/336_09/papers/Sheri.pdf However, I am getting confused with definitions 3.3 and 3.4. I am not sure ...
0
votes
1answer
771 views

Field extension, primitive root of unity

Let $\xi_5$ be a primitive fifth root of unity in $\mathbb{C}$, then we know the extension $\mathbb{Q}(\xi_5)\supseteq\mathbb{Q}$ has degree 4 so the Galois group is of order 4. I am trying to find ...
4
votes
2answers
236 views

Math and Origami

I am working on a project for class about the mathematics behind origami and write now I am looking into what is and is not constructible. I've gotten to the definition of origami constructible points ...
2
votes
2answers
818 views

Is Complex Numbers the biggest field? If yes, is there any easy proof to understand it?

Is the Complex Numbers the biggest field? If yes, does anyone have a "simple"/"easy to understand" proof?
1
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0answers
46 views

How do I prove the multiplicativity of separable degree in general?

Let $K/E/F$ be a tower of algebraic extensions. How do I prove that $[K:E]_s[E:F]_s = [K:F]_s$? This is done in all the books I searched for finite extensions (when it follows trivially from a ...
2
votes
1answer
156 views

Basis of $\Bbb Q(\sqrt[n]7)$ over $\mathbb Q$

I have the field extension $\Bbb Q(\sqrt[n]7)/\mathbb Q$ for $n\ge2$. I need to find the basis for this field extension but I'm not sure what it is.
0
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1answer
43 views

Subring of $\Bbb F_{125}$ generated by ${1}$

I have a field $\Bbb F_{125}$ which is $$\Bbb F_{125}=\frac{\Bbb F_5[X]}{\langle X^3+X+1\rangle}$$ I have been asked to find the subring generated by the multiplicative identity of this field. I ...
3
votes
1answer
211 views

Show that $\overline {\mathbb Q(x)}$ is isomorphic to a subfield of $\mathbb C$

Show that $\overline {\mathbb Q(x)}$ is isomorphic to a subfield of $\mathbb C$. Here $\mathbb Q(x)$ is the field of rational functions (the field of fractions of the polynomial ring $\mathbb ...
1
vote
1answer
153 views

extending an isomorphism of a field to an embedding

Let $K$ be an algebraic extension of $F$, contained in an algebraic closure $E$ of $F$. Let $\alpha \in K$, $m(x)$ be its minimal polynomial over $F$ and $\beta$ a root of $m(x)$ in $E$, then there ...