Wikipedia:Reference desk/Archives/Mathematics/2012 August 5

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August 5

Plane

a) How many planes can be made to pass through-

1. a line and a point not on the line?
2. two points?
3. three distinct points?

b) In the above question, what is the meaning of "plane"? Sunny Singh (DAV) (talk) 07:41, 5 August 2012 (UTC)

This looks like homework, so I won't just give you the answer. If you want to list your answers, we will review them for you. See plane (geometry). StuRat (talk) 07:46, 5 August 2012 (UTC)

Is that the exact wording of the question? It is a little strange - at least one of the answers (I won't say which or how many) is "infinitely many", which is an odd answer. Perhaps the question is supposed to mean how many parameters does the family have? --Tango (talk) 12:35, 5 August 2012 (UTC)

Infinity is neither odd nor even! —Tamfang (talk) 05:37, 6 August 2012 (UTC)

Or is it both? --Tango (talk) 22:30, 6 August 2012 (UTC)

That's what I thought at first, but there's no integer n for which infinity is either 2n or 2n+1, so it really is neither. Infinity itself is of course not an integer. Though if we extend to ordinals it becomes a matter of the order of operands. -- Meni Rosenfeld (talk) 18:42, 9 August 2012 (UTC)

Meaning of a symbol

In

Silverman, J. H. (1988), "Wieferich's criterion and the abc-conjecture", Journal of Number Theory, 30 (2): 226–237, doi:10.1016/0022-314X(88)90019-4

on page 227, Silverman states Theorem 1. In this theorem, he uses a symbol the meaning of which is not clear to me. I am talking about the symbol between the second vertical bar and log(X). Unfortunately I can reproduce that symbol here using neither a character nor TeX, so I provide the following screenshot (click on the image to enlarge).

I am talking about the circled symbol. What does it mean? -- Toshio Yamaguchi (tlk−ctb) 08:20, 5 August 2012 (UTC)

For reference, this appears to be the Unicode Character 'DOUBLE NESTED GREATER-THAN' (U+2AA2) under Supplemental Mathematical Operators with an underset Greek lower-case alpha character. I hope this saves others the trouble of looking it up. — Quondum^☏ 11:23, 5 August 2012 (UTC)

I believe this is Vinogradov notation, indicating an inequality up to a constant only involving α. Sławomir Biały (talk) 15:50, 5 August 2012 (UTC)

Question about the meaning of another symbol

In Granville, A.; Monagan, M. B. (1988), "The First Case of Fermat's Last Theorem is true for all prime exponents up to 714,591,416,091,389", Transactions of the American Mathematical Society, 306 (1): 329–359, doi:10.1090/S0002-9947-1988-0927694-5. on page 332, what does Q(ξ) stand for? I guess that Q denotes the set of rational numbers and ξ is defined as: cos 2π/p + i sin 2π/p. Still I am not sure I understand what exactly Q(ξ) is. Can someone shed some light on this? -- Toshio Yamaguchi (tlk−ctb) 14:40, 5 August 2012 (UTC)

It is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers. This is known as a number field. In this particular case (where ξ is a root of unity), the field is a cyclotomic number field. Sławomir Biały (talk) 15:58, 5 August 2012 (UTC)

So if I understand this correctly, Q(ξ), to express it simply, contains all rational numbers plus (from Field (mathematics)#Definition and illustration) a multiplicative inverse for every nonzero element. Per algebraic number, such a number is a root of a polynomial in one variable which is cos 2π/p + i sin 2π/p here. So Q(ξ) is the rational numbers plus their multiplicative inverses plus all roots of cos 2π/p + i sin 2π/p. Is that correct? -- Toshio Yamaguchi (tlk−ctb) 17:52, 5 August 2012 (UTC)

It's actually just all polynomials of the form

$a_{p-2}\xi ^{p-2}+a_{p-3}\xi ^{p-3}+\cdots +a_{0}$

with the $a_{i}$ rational. There is no need to include additional multiplicative inverses, since the multiplicative inverses can also always be put in this form. Sławomir Biały (talk) 18:07, 5 August 2012 (UTC)

Is it synonymous to say Q(ξ) is the closure of the set Q∪{ξ} under addition and multiplication? —Tamfang (talk) 05:36, 6 August 2012 (UTC)

Yes. Saying that you don't need multiplicative inverses too, as Sławomir Biały notes, is precisely the same thing as saying that ξ is algebraic over Q, saying that it is an Algebraic number. That wouldn't be true if you were adjoining an indeterminate "x" or a transcendental number, say, π .John Z (talk) 21:40, 7 August 2012 (UTC)

It's not clear to me that every algebraic ξ would makes multiplicative inverses redundant; not all algebraic numbers have an inverse as convenient as ξ^p-1. —Tamfang (talk) 22:28, 8 August 2012 (UTC)

Take a minimal polynomial for ξ. For x= ξ, it is zero. Move the constant term to the other side, and factor out ξ. This gives ξ^-1 as a polynomial in ξ. John Z (talk) 00:29, 9 August 2012 (UTC)

Two questions (not homework)

1. Is the Matrix invertible?
2. Are the rings of power commutative rings?

Thanks — Preceding unsigned comment added by פדיחה (talk • contribs) 16:50, 5 August 2012 (UTC)

1. No, the Matrix is not invertible, because it has a kernel.
2. No, a ring of power is not commutative, because the center of the ring is not the entire ring. --COVIZAPIBETEFOKY (talk) 17:25, 5 August 2012 (UTC)

It's times like this I wish I could "like" Wikipedia comments. —Anonymous Dissident^Talk 14:28, 10 August 2012 (UTC)

Like Joefromrandb (talk) 20:32, 10 August 2012 (UTC)