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Sum of three squares
perfect_radio 9
N
by RobertRogo
Source: RMO 2004, Grade 12, Problem 4
Let 
be a field of characteristic 
, 
.
(a) Prove that
is the square of an element from 
(b) Prove that any element
from can be written as the sum of three squares, each , of elements from .
(c) Can
be written in the same way?
Marian Andronache
be a field of characteristic 
, 
.(a) Prove that
is the square of an element from 
(b) Prove that any element
from can be written as the sum of three squares, each , of elements from .(c) Can
be written in the same way?Marian Andronache
9 replies
|A/pA|<=p, finite index=> isomorphism - OIMU 2008 Problem 7
Jorge Miranda 2
N
by pi_quadrat_sechstel
Let 
be an abelian additive group such that all nonzero elements have infinite order and for each prime number we have the inequality 
, where 
, 
(where the sum has summands) and 
is the order of the quotient group 
(the index of the subgroup 
).
Prove that each subgroup of of finite index is isomorphic to .
be an abelian additive group such that all nonzero elements have infinite order and for each prime number we have the inequality 
, where 
, 
(where the sum has summands) and 
is the order of the quotient group 
(the index of the subgroup 
).Prove that each subgroup of
of finite index is isomorphic to .
2 replies
Divergent sum of inverses
toto1234567890 7
N
by GreenKeeper
Source: Miklós Schweitzer 2014, P5
Let 
be a non-real algebraic integer of degree two, and let 
be the set of irreducible elements of the ring ![$ \mathbb{Z}[ \alpha] $](//latex.artofproblemsolving.com/3/1/6/316d00f1eaea5f28346a44deedb58bc83f41657e.png)
. Prove that
![\[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]](//latex.artofproblemsolving.com/d/3/d/d3d48d0ac7bee298334a11a14559e3b05ab0cd1d.png)
be a non-real algebraic integer of degree two, and let 
be the set of irreducible elements of the ring ![$ \mathbb{Z}[ \alpha] $](http://latex.artofproblemsolving.com/3/1/6/316d00f1eaea5f28346a44deedb58bc83f41657e.png)
. Prove that![\[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]](http://latex.artofproblemsolving.com/d/3/d/d3d48d0ac7bee298334a11a14559e3b05ab0cd1d.png)
7 replies
Miklos Schweitzer 1975_3
ehsan2004 1
N
by FFA21
Source: semigroup without proper two-sided ideals
Let 
be a semigroup without proper two-sided ideals and suppose that for every 
at least one of the products 
and 
is equal to one of the elements 
. Prove that either 
for all or 
for all .
L. Megyesi
be a semigroup without proper two-sided ideals and suppose that for every 
at least one of the products 
and 
is equal to one of the elements 
. Prove that either 
for all or 
for all .L. Megyesi
1 reply
Miklos Schweitzer 1967_3
ehsan2004 2
N
by FFA21
Prove that if an infinite, noncommutative group 
contains a proper normal subgroup with a commutative factor group, then also contains an infinite proper normal subgroup.
B. Csakany
contains a proper normal subgroup with a commutative factor group, then also contains an infinite proper normal subgroup.B. Csakany
2 replies
Romania National Olympiad ,2013,problem 2,grade 12
ionbursuc 5
N
by AndreiVila
Given a ring 
that meets both of the following conditions:
(1) is not a field, and
(2) For every non-invertible element
of 
, there is an integer 
(depending on ) such that 
.
Show that
(a)
for every 
, and
(b)
for every non-invertible 
.
that meets both of the following conditions:(1)
is not a field, and(2) For every non-invertible element
of 
, there is an integer 
(depending on ) such that 
.Show that
(a)
for every 
, and(b)
for every non-invertible 
.
5 replies
A Strange Generalization of Normal Subgroups
swizzlestick3 1
N
by swizzlestick3
Source: My Own.
I've noticed an interest in group theory stuff on here. Hence, I decided to post this problem which I came up with a while back. The question is very open ended, so here goes.
Let
be a finite group. A subgroup 
of is called normal-ish in if the conjugates of in exactly cover a subgroup of . That is, if
is a subgroup of. Investigate this notion. I have some partial results which you can DM me for if you'd like. Cheers! :)
Let
be a finite group. A subgroup 
of is called normal-ish in if the conjugates of in exactly cover a subgroup of . That is, if
is a subgroup of
. Investigate this notion. I have some partial results which you can DM me for if you'd like. Cheers! :)
1 reply
Define the polynomial f_a
WakeUp 3
N
by KevinYang2.71
Let 
be a field having 
elements, where is a prime and 
is an arbitrary integer number. For any 
, one defines the polynomial 
. Show that:
is divisible by 
;
has at least 
essentially different irreducible factors ![$K[X]$](//latex.artofproblemsolving.com/b/8/f/b8f4ddba89151fa31ddf44f6c22015162f2b58b0.png)
.
be a field having 
elements, where is a prime and 
is an arbitrary integer number. For any 
, one defines the polynomial 
. Show that:
is divisible by 
;
has at least 
essentially different irreducible factors ![$K[X]$](http://latex.artofproblemsolving.com/b/8/f/b8f4ddba89151fa31ddf44f6c22015162f2b58b0.png)
.
3 replies
Counting solutions to exponents under group homomorphism
R.e.d.a.c.t.e.d 0
Given 
, a group homomorphism, Where 
are finite groups and Let 
be a fixed positive integer. Then prove that 
, a group homomorphism, Where 
are finite groups and Let 
be a fixed positive integer. Then prove that 
0 replies
Any commutatuve subring of A is a field implies A is a field
WakeUp 10
N
by KevinYang2.71
Let be a ring.
Show that the set 
is a subring of the ring .
Prove that, if any commutative subring of is a field, then is a field.
be a ring. Show that the set 
is a subring of the ring . Prove that, if any commutative subring of is a field, then is a field.
10 replies
