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
26 views

Field of prime characteristic over two indeterminates

Let $F$ have prime characteristic $p$ and let $E = F(Y,Z)$, where $Y, Z$ are indeterminates. Let $L=F(Y^{p} , Z^{p})$ $\subseteq E$. a. Show that $\alpha^{p} \in L$ for all $\alpha \in E$. b. Show ...
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0answers
80 views

In which Fields, does $x^n-x$ have a multiple zero?

In which Fields, does $x^n-x$ have a multiple zero? Attempt: Let $f(x) = x^n-x$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. An irreducible ...
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0answers
17 views

Ordered field - Adding inequalities (Looking for a proof)

In an ordered field, it is allowed to add two inequalities. Where can I find a detailed proof?
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2answers
48 views

What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
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1answer
37 views

Proving that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$

if $f(x)$ is a cubic irreducible polynomial over $\mathbb Z_3$, prove that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$ Attempt: $f(x) = \alpha ...
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0answers
27 views

Presence of non square elements in $GF(p)$

I came across a problem which had a line of explanation as follows : Let $a \in GF(p). a$ is a non square in $GF(p),~ p \neq 2 \implies \nexists~ b\in GF(p)~~|~~a =b^2 $. But, is it really possible ...
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1answer
48 views

Counting the roots of a polynomial over a finite field

Let $\mathbb{F}_{11}$ be the field of 11 elements and let $\mathcal{K}$ be the splitting field of $x^{3} - 1$ over $\mathbb{F}_{11}$. How many roots does $(x^{2} - 3)(x^{3} - 3)$ have in ...
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0answers
22 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
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1answer
26 views

Some questions about sub-fields of the field of complex numbers

Given a sub-field $f$ of the field $\mathbb{C}$ of complex numbers, is there a name for the smallest sub-field $F(f)$ of $\mathbb{C}$ such that (1) $F(f)$ contains $f$ as a sub-field and (2) ...
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0answers
20 views

determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
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0answers
34 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
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3answers
92 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
1
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1answer
29 views

A field of characteristic zero is perfect

How do you prove that a field F of characteristic zero is perfect, or rather that every irreducible f(x) in F[x] is separable? Thank you!
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0answers
11 views

intersection is isomorphic to $Aut(F)$

Prove that for any finite field $F$ there exist two subgroups of $GL(|F|,\mathbb{R})$ whose intersection is isomorphic to $Aut(F)$
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1answer
24 views

Field automorphisms and fixed fields

I currently have to cope with field automorphisms. I already understood that any field automorphism of $\mathbb{C}$ must fix all elements in $\mathbb{Q}$. My question is the following: Assume a ...
5
votes
2answers
92 views
+100

Generators of the Relations of a Galois Extension

Let $K$ be a Galois extension of $\mathbb{Q}$ of degree $n$. Pick some primitive element and take the roots $a_1, ..., a_n$ of its minimal polynomial. Then the evaluation map $\mathbb{Q}[x_1, ..., ...
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0answers
67 views

What is the Difference Between an Algebra and a Field?

The Wikipedia page for $\sigma$-algebra, as well as some other resources I'm studying, say this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on ...
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2answers
42 views

Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$

Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is ...
3
votes
2answers
141 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
2
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1answer
38 views

Extending Homomorphism into Algebraically Closed Field

If we are given a homomorphism g between a field k and an algebraically closed field $\Omega$, and a field k' which is a finite algebraic extension of k, how do we extend g to a homomorphism g' from ...
4
votes
1answer
35 views

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$. Attempt: $F \subset F(a^3) \subseteq ...
2
votes
1answer
22 views

A field extension of prime degree

Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$ Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. ...
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2answers
25 views

existence of some algebraic closure

I was curious somethings when I studied algebraic closure in Hungerford's algebra. I know that every field has an algebraic closure. But I have the following question. Let K be a field and F an ...
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2answers
48 views

What are subfields of $\mathbb{C}$?

I only took the first undergraduate abstract algebra course, so i don't know (at all) what Galois theory is about. I'm asking this question since i'm not sure of the definition of inner product space ...
2
votes
1answer
30 views

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f(x)$ and $\deg g(x)$ are relatively prime.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\gcd(~\deg g(x),\deg f(x)~)=1$. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$ ...
3
votes
1answer
29 views

Field theorem: impossible to satisfy three equations simultaneously in an integer field?

Suppose $a_0,a_1,b,c,d$ with $a_0 \neq a_1$ are integers, or elements of any ${\mathbb Z}_p$ for prime $p$. A while ago I was attempting to find a set of such numbers that satisfied the equations ...
3
votes
1answer
40 views

Question about calculating the degree of a finite field extension.

I have a question about calculating the degree of a finite field extension over $\mathbb{Q}$. This is problem 18 in chapter 1 of Patrick Morandi's Field and Galois theory. The problem asked to show ...
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0answers
38 views

Why do nth roots (radicals) have closed forms whilst other polynomial roots do not?

The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class ...
7
votes
1answer
72 views

Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
2
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0answers
56 views

Question on a finite field extension of $\mathbb{Q}$

I have a polynomial $p(x) \in \mathbb{Q}[x]$ and is irreducible over $\mathbb{Q}$. Let it be of degree $n$ and $\alpha_1, ..., \alpha_n$ be its roots. I know that $$ \mathbb{Q}(\alpha_i) \cong ...
4
votes
2answers
35 views

Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
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0answers
7 views

Solutions to a polynomial equation in a PAC field not lying in a subfield

Suppose $f(x,y)$ is an absolutely irreducible polynomial over a PAC (pseudo algebraically closed) field $K$ such that $x,y$ actually appear in $f$. Let $L$ be a proper subfield of $K$. Are we ...
3
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3answers
92 views

Finitely many embeddings of a finite extension in an algebraic closure

So I'm reading through Lang's Algebra, and he keeps saying something along the following lines: "Let $K$ be a finite extension of a field $k$ and let $\sigma_1,\ldots,\sigma_r$ be the distinct ...
3
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0answers
45 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field ...
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0answers
33 views

A question regarding a lemma in Perrin's Algebraic Geometry.

Algebraic Geometry by Perrin says the following: Let $k$ be an uncountable algebraically closed field and let $K$ be an extension of $k$ whose dimension is at most countable. Then $K=k$. He ...
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1answer
37 views

Is the field formed by algebraic elements of an extension field over $F$ isomorphic to $F[t]$?

Say $K/F$ is a field extension. The elements in $K$ that are algebraic over $F$ form a subfield of $K$. Is this subfield isomorphic to $F[t]$? What would this isomorphism look like? This is not a ...
1
vote
1answer
35 views

Prove that any two bases of a field extension have the same cardinality.

Suppose $E$ and $F$ are subfields of $\mathbb{R}$ with $F\subseteq E$. Prove that any two bases of $E/F$ have the same cardinality. The definition of a basis I am using is any finite set $S\subseteq ...
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2answers
50 views

There always exists a subfield of $\mathbb C$ which is a splitting field for $f(x)$ $\in$ $Q[X]$?

So I've been studying field theory on my own, and I just started learning about splitting fields. Based on my understandings if a polynomial, $f(x)$ $\in$ $Q[X]$, then there should be always a ...
2
votes
2answers
29 views

Transitivity Property of Separable Extensions

I was looking for some proof for the transitivity property of separable field extensions. Although this might sound like a very well-known fact and is referred to frequently, I do not seem to find a ...
1
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3answers
27 views

Quadratic number fields containing primitive roots of unity

A problem from Michael Artin's Algebra (Second Edition) from Fields: Determine the quadratic number fields $\mathbb{Q}[\sqrt{d}]$ that contain a primitive $n$th root of unity, for some integer $n$. ...
2
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2answers
74 views

How do elements of $\mathbb{R}(xy,x+y)$ look like?

I have problems with determining what are typical elements of such field $\mathbb{R}(xy,x+y)$ In one indeterminate it is easier as $\mathbb{R}(x)=\Bigl\{\frac{f(x)}{g(x)}, g(x)\neq 0, ...
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1answer
47 views

Describe the elements in $Q(\pi)$

Describe the elements in $Q(\pi)$ Attempt: $Q(\pi)$ is the smallest field which contains $Q$ and $\pi$ We know that $\nexists~ f(x) \in Q[x]$ such that $f(\pi)=0$ Hence, $Q[x]/\langle p(x) ...
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0answers
14 views

Is there a direct way of proving that all splitting fields are isomorphic?

Given $f(x)\in F[X]$, let $E,E'$ be two extension fields of $f$ over $F$, then $E \approx E'$. Now, I've seen a proof involving directly constructing an isomorphism, but I'm searching for another ...
1
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1answer
34 views

Every field with characterisitic $p$ contains the field $\mathbb{Z}_p$

I seem to hold a very loose grasp of the concept of fields - I've encountered this question: Show that every finite field with characteristic $p$ contains $\mathbb{Z}_p$ (i.e. $\mathbb{Z}_p = ...
0
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2answers
30 views

Zeroes of f(x) in a splitting field $E $ have the same multiplicity

Let $f(x)$ be an irreducible polynomial over a field $F$ and let $E$ be a splitting field of $f(x)$ over $F$. Then all the zeroes of $f(x)$ in $E$ have the same multiplicity. The proof of this ...
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1answer
52 views

Are order isomorphic real closed fields isomorphic?

There are counterexamples to order isomorphisms of ordered fields being field isomorphisms, see Is the multiplicative structure of a totally ordered field unique?. However, Wikipedia suggests that for ...
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2answers
33 views

Discrete valuations of the rational numbers

I'm trying to find every discrete valuation on the field of rational numbers. If $a\in \mathbb Q$, we can write $a=p^j\frac{x}{y}$, where $p$ is a prime number and $p\nmid x$ and $p\nmid y$. We can ...
2
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1answer
36 views

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$.

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$. I have some queries regarding this theorem of existence of ...
2
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3answers
134 views

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...
2
votes
1answer
41 views

Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...