Quadratic forms
Indecomposable integers
Biquadratic elds

Indecomposable integers and universal quadraticforms
Magdaléna Tinková(with M. ech, D. Lachman, J. Svoboda, and K. Zemková)
Department of Algebra, Faculty of Mathematics and Physics,
Charles University, Prague

This work was supported by the Charles University, project GA UK No. 1298218.

May 23, 2018

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Outline1

Quadratic forms

2

Indecomposable integers

3Biquadratic elds

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

K = Q(β)  totally real number eldβ  root of polynomial of degree dβ1 = β, β2 , . . . , βd  real conjugates of βα ∈ K ⇒ α = a0 + a1 β + a2 β 2 + · · · + ad−1 β d−1
α is totally positive if
α =a0 + a1 β + a2 β 2 + · · · + ad−1 β d−1 > 0,
α2 =a0 + a1 β2 + a2 β22 + · · · + ad−1 β2d−1 > 0,..
.
αd =a0 + a1 βd + a2 βd2 + · · · + ad−1 βdd−1 > 0
OK  ring of algebraic integers of K+OK
 totally positive elements of OK
M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Quadratic forms
Denition
Quadratic form Q over K with variables x1 , x2 , . . . , xn is anexpression
XQ(x1 , x2 , . . . , xn ) =aij xi xj
1≤i≤j≤nwhere aij ∈ OK .
ExampleSum of squares

Pn2i=1 xi .

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Our quadratic form istotally positive denite if Q(γ1 , . . . , γn ) is totally positive forall γj ∈ OK , not all zero;classical if 2|aij for all i 6= j ;+universal if it represents all elements of OK, i.e., for every+α ∈ OK, there exist γ1 , . . . , γn ∈ OK such thatX

α=aij γi γj1≤i≤j≤nAssumption: Our forms are totally positive denite and classical.
M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Question: When are our forms universal?
Theorem (Lagrange, 1770; Maas, 1941)

The sum of four squares is universal
over Q. The sum of three√squares is universal over Q( 5).
Theorem (Siegel, 1945)

The√sum of any number of squares is universal only over Q andQ( 5).

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Conjecture (Kitaoka)
There are only nitely many elds which admit ternary universal
quadratic forms.
Theorem (Chan, Kim, Raghavan, 1996)

The only quadraticnumberelds whichadmit ternary universal√√√forms are Q( 2), Q( 3), and Q( 5).
Theorem (Blomer, Kala, 2015; Kala, 2016)

For each M , there are innitely many real quadratic elds that do
not admit universal forms with M variables.

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic eldsIndecomposable integers

Denition+α ∈ OK
is indecomposable if it cannot be expressed as α = β + γ+
where β, γ ∈ OK.

indecomposable integers are dicult to represent by quadraticforms
we often use them as coecients of our forms

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Quadratic case√
K = Q( p), p squarefree positive rational integer√√p+1
OK = Z[ p] or OK = Z[ 2 ] for p ≡ 1 (mod 4)√√p−1[u0 , u1 , . . . , us ] = continued fraction of p or 2
p−1 = 1, p0 = u0 , pn+1 = pn−1 + un+1 pn
q−1 = 0, q0 = 1, qn+1 = qn−1 + un+1 qn
pn
qn = [u0 , u1 , . . . , un ]M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

DenitionSet√
αn = pn + qn p,αn,r = αn + rαn+1
where 0 ≤ r ≤ un+2 .
Elements αn are called convergents.
Theorem (Perron, 1913; Dress, Scharlau, 1982)

The elements αn,r where n is odd and 0 ≤ r ≤ un+2 are all√indecomposable integers in Q( p).
We do not have such a characterization of indecomposable integers
in elds of higher degrees.
M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Biquadratic elds

√ √K = Q( p, q), p, q squarefree positive rational integers,
p 6= qpqr = gcd(p,q)
2√√√
K has quadratic subelds Q( p), Q( q) and Q( r)Our questions:
1

Which universal quadratic forms have we in biquadratic elds?

2

Do indecomposable integers from quadratic subelds remain
indecomposable in our biquadratic eld?

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Ring of integers of K
Theorem (Williams, 1970)

The only possible integral bases for biquadratic elds are123n √ √ √ √ op+ r
1, p, q, 2,n √√ √ √ o1+ qp+ r
1, p, 2 , 2,n√√√ √ √ o1+ p 1+ q 1± p+ q+ r
1, 2 , 2 ,.
2Theorem (ech, Lachman, Svoboda, T., Zemková, 2018)√In cases 1. and 2., indecomposables from Q( q) are√ √indecomposable in Q( p, q).

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Theorem (ech, Lachman, Svoboda, T., Zemková, 2018)√√
p−1
If p < r, then convergents of p (resp. 2 ) remain√ √indecomposable in Q( p, q).
Theorem (ech, Lachman, Svoboda, T., Zemková, 2018)√√√
If p < Mp r, then indecomposables of Q( p) remain√ √indecomposable in Q( p, q).

Open question: Do indecomposable integers from quadraticsubelds always remain indecomposable in our biquadratic eld?

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Universal forms over biquadratic elds

Theorem (ech, Lachman, Svoboda, T., Zemková, 2018)√ √Any universal√ √form over Q( 2, 3) must have at least 5 variables;
over Q( 6, 19) it must have at least 6 variables.

M. Tinková et al.

Indecomposable integers and universal quadratic forms

Quadratic forms
Indecomposable integers
Biquadratic elds

Thank you for your attention.

M. Tinková et al.

Indecomposable integers and universal quadratic forms