Abstract algebra questions, solved and explained

Recent questions in Abstract algebra

foass77W 2021-01-08 Answered

Find the inverse of $x + 1 \in Q \frac{x}{x^{3} - 2}$ . Explain why this is the same as finding the inverse of $\sqrt[3]{2} \in R$ .

Lewis Harvey 2021-01-06 Answered

Prove that a group of even order must have an element of order 2.

Joni Kenny 2021-01-05 Answered

Let a,b be coprime integers. Prove that every integer $x > a b - a - b$ can be written as $n a + m b$ where n,m are non negative integers. Prove that $a b - a - b$ cannot be expressed in this form.

Dolly Robinson 2021-01-04 Answered

Let F be a field and consider the ring of polynominals in two variables over F,F[x,y]. Prove that the functions sending a polyomial f(x,y) to its degree in x, its degree in y, and its total degree (i.e, the highest $i + j$ where $x^{i} y^{i}$ appears with a nonzero coefficient) all fail o be norm making F[x,y] a Euclidean domain.

texelaare 2020-12-30 Answered

In the froup $Z_{12}$ , find $| a |, | b |$ , and $| a + b |$
$a = 3, b = 8$

hexacordoK 2020-12-27 Answered

Use Principle of MI to verify
(i) If $n \in Z$ is a positive ineger then $2^{n} 3^{x n} - 1$ divisible by 17.
(ii) For all positive integers $n \geq 5$ ,
$2^{k} > k^{2}$

Rui Baldwin 2020-12-15 Answered

Let F be i field with subfields K,L. Prove that there is a largest subfield of F contained in both K and L, and a smallest subfield of containung both K and L.

nicekikah 2020-12-15 Answered

How many subgroups of order 4 does $D_{4}$ have?

Harlen Pritchard 2020-12-15 Answered

Suppose G is a group, H a subgroup of G, and a and b elements of G. If $a \in b H$ then $b \in a H$ .

iohanetc 2020-11-29 Answered

How to find the value of U(n) on abstract algebra
Example U(8) = {1,3,5,7}

Khadija Wells 2020-11-20 Answered

Prove that in any group, an element and its inverse have the same order.

mattgondek4 2020-11-11 Answered

An nth root of unity epsilon is an element such that $ϵ^{n} = 1$ . We say that epsilon is primitive if every nth root of unity is $ϵ^{k}$ for some k. Show that there are primitive nth roots of unity $ϵ_{n} \in C$ for all n, and find the degree of $Q \to Q (ϵ_{n})$ for $1 \leq n \leq 6$

rocedwrp 2020-11-10 Answered

Find th eminimal polynomial of $\sqrt{2} + \sqrt{3}$ over $Q$

Mylo O'Moore 2020-10-23 Answered

Write down a definition of a subfield. Prove that the intersection of a set of subfields of a field F is again a field

Isa Trevino 2020-10-23 Answered

Let F be a field. Prove that there are infinitely many irreducible monic polynomials

naivlingr 2020-10-20 Answered

Let $p, q \in Z$ be district primes. Prove that $Q (\sqrt{p}, \sqrt{q}) = Q (\sqrt{p} + \sqrt{q})$ , and that $Q \subseteq Q (\sqrt{p} + \sqrt{q})$ is a degree 4 extension.

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Algebra