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July 23

Is every infinite field with cardinality aleph-0 isomorphic to the rationals?

Just wondering. --2404:2000:2000:5:0:0:0:C2 (talk) 00:31, 23 July 2014 (UTC)[reply]

No. Consider the field Q(sqrt(2)) (I'm no longer sure of the notation; what I mean is the smallest field containing the rationals and sqrt(2)). Any isomorphism between them would have to fix the rationals, so there's nowhere for sqrt(2) to go on the Q side. --Trovatore (talk) 00:47, 23 July 2014 (UTC)[reply]

Your notation is pretty standard, sometimes rendered with the "blackboard bold" or square brackets, e.g. as

\mathbb {Q} [{\sqrt {2}}]

. Our relevant article is Algebraic_number_field. SemanticMantis (talk) 15:52, 23 July 2014 (UTC)[reply]

I think I learned square brackets for the extension as a ring, parentheses (round brackets) for the field. --Trovatore (talk) 17:59, 23 July 2014 (UTC)[reply]

That is also what I learned. However,

\mathbb {Q} [{\sqrt {2}}]=\mathbb {Q} ({\sqrt {2}})

, so it is not very important in this case. —Kusma (t·c) 18:02, 23 July 2014 (UTC)[reply]

Even worse: the algebraic closure of a finite field is countable but doesn't even contain an isomorphic image of the rationals. —Kusma (t·c) 07:29, 23 July 2014 (UTC)[reply]

But any field of characteristic 0 contains the rational as a subfield --77.126.41.10 (talk) 14:53, 24 July 2014 (UTC)[reply]

equation

I'm trying to figure out a way to equitably pay off debt with my spouse. She makes 71% of what I make.

Let's say debt d=1750 and d=m+h (mine and her contribution)
Does h=0.71m??
And then does d=0.71m+m??
Does m=1232.39?

I'm doubting because then h=517.61 and I think h/m=0.71, but it doesn't...

I want to ultimately make an excel spreadsheet. Thanks

You're right up to d=0.71m+m, but you must have made a mistake after that, because it gives m = d/1.71 = 1023.39, and so h = 726.61. AndrewWTaylor (talk) 16:34, 23 July 2014 (UTC)[reply]

Wow, thanks for the quick answer. That makes sense. Maybe you could help me spot my mistake, too?

d=0.71m+m
Divide d by 0.71 and cancel from other side
so 2464.79=2m
m=1232.39

Step 2, somewhere I think. Thank you again! — Preceding unsigned comment added by 166.147.123.165 (talk) 16:53, 23 July 2014 (UTC)[reply]

If you want a simple formula for a spreadsheet for proportionally dividing a quantity, think in terms of each income as a fraction of the total income. So, say you each earn a = $1 and b = $0.71 respectively. Then an apportionment multipliers would be a′ = a/(a+b) = 0.585 and b′ = b/(a+b) = 0.415, so m = a′d and h = b′d, and these factors a′ and b′ can be reused for splitting other amounts. —Quondum 17:54, 23 July 2014 (UTC)[reply]

And as to the specific algebraic error, yes, step 2. You can divide both sides of an equation by 0.71, but the right hand side would become (0.71m + m)/0.71 = m + m/0.71, not m + m. What you should have done is to see that 0.71m + m = 1.71m, and then divide both sides by 1.71. -- ToE 21:21, 23 July 2014 (UTC)[reply]

This is how you should solve the problem. Very simple ratio.

You should think like this

For every $100 I earn, she earns $71
Thus we earn a total of $171 dollars ($100 + $71) for every $100 that I earn
Thus my ratio of the debt is 100/(100+71)
Thus her ratio of the debt is 71/(100+71)

d = 1750
m = 100/(100+71) * d
h = 71/(100+71) * d

Very easy. You don't need an excel spreadsheet. You just need to think clearly. 202.177.218.59 (talk) 04:12, 24 July 2014 (UTC)[reply]

I'm not at all certain why the paying off of the debt is to be divided up according to how much each earns. There are lots of ideas on how it should be done, e.g. see fair division and airport problem and see how complex and confused these sorts of things can be made at Entitlement (fair division)#Entitlement in the Talmud. Basically you sre agreeing with proportional tax rather than a progressive tax for income. Dmcq (talk) 11:24, 25 July 2014 (UTC)[reply]