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On the product of vector spaces in a commutative field extension
Let K ⊂ L be a commutative field extension. Given K-subspaces A , B of L, we consider the subspace 〈 A B 〉 spanned by the product set A B = { a b | a ∈ A , b ∈ B } . If dim K A = r and dim K B = s , how small can the dimension of 〈 A B 〉 be? In this paper we give a complete answer to this question i...
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| Published in: | Journal of number theory 2009-02, Vol.129 (2), p.339-348 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c374t-e2ac66d7d8cf693aa015d02703e335fc3385a5fdffa10b008bbd62878b9b4b6e3 |
| container_end_page | 348 |
| container_issue | 2 |
| container_start_page | 339 |
| container_title | Journal of number theory |
| container_volume | 129 |
| creator | Eliahou, Shalom Kervaire, Michel Lecouvey, Cédric |
| description | Let
K
⊂
L
be a commutative field extension. Given
K-subspaces
A
,
B
of
L, we consider the subspace
〈
A
B
〉
spanned by the product set
A
B
=
{
a
b
|
a
∈
A
,
b
∈
B
}
. If
dim
K
A
=
r
and
dim
K
B
=
s
, how small can the dimension of
〈
A
B
〉
be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on
dim
K
〈
A
B
〉
turns out, in this case, to be provided by the numerical function
κ
K
,
L
(
r
,
s
)
=
min
h
(
⌈
r
/
h
⌉
+
⌈
s
/
h
⌉
−
1
)
h
,
where
h runs over the set of
K-dimensions of all finite-dimensional intermediate fields
K
⊂
H
⊂
L
. This bound is closely related to one appearing in additive number theory. |
| doi_str_mv | 10.1016/j.jnt.2008.06.004 |
| format | article |
| fullrecord | <record><control><sourceid>hal_cross</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_00259373v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0022314X0800142X</els_id><sourcerecordid>oai_HAL_hal_00259373v1</sourcerecordid><originalsourceid>FETCH-LOGICAL-c374t-e2ac66d7d8cf693aa015d02703e335fc3385a5fdffa10b008bbd62878b9b4b6e3</originalsourceid><addsrcrecordid>eNp9kEtLAzEUhYMoWB8_wF22Lma8mUwyM7ixFLVCoRsFdyGTB83QTkqSDvrvTam4dHXhcr4D50PojkBJgPCHoRzGVFYAbQm8BKjP0IxAxwvCWXuOZgBVVVBSf16iqxgHAEJYw2boaT3itDF4H7w-qIS9xZNRyQcc91KZiN2IJVZ-tzskmdxksHVmq7H5SmaMzo836MLKbTS3v_cafbw8vy-WxWr9-raYrwpFmzoVppKKc93oVlneUSmBMA1VA9RQyqyitGWSWW2tJNDnGX2vedU2bd_1dc8NvUb3p96N3Ip9cDsZvoWXTiznK3H85YWsow2dSM6SU1YFH2Mw9g8gII66xCCyLnHUJYBntM7M44kxecTkTBBROTMqo13IQoT27h_6B79Ycm0</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>On the product of vector spaces in a commutative field extension</title><source>Elsevier:Jisc Collections:Elsevier Read and Publish Agreement 2022-2024:Freedom Collection (Reading list)</source><creator>Eliahou, Shalom ; Kervaire, Michel ; Lecouvey, Cédric</creator><creatorcontrib>Eliahou, Shalom ; Kervaire, Michel ; Lecouvey, Cédric</creatorcontrib><description>Let
K
⊂
L
be a commutative field extension. Given
K-subspaces
A
,
B
of
L, we consider the subspace
〈
A
B
〉
spanned by the product set
A
B
=
{
a
b
|
a
∈
A
,
b
∈
B
}
. If
dim
K
A
=
r
and
dim
K
B
=
s
, how small can the dimension of
〈
A
B
〉
be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on
dim
K
〈
A
B
〉
turns out, in this case, to be provided by the numerical function
κ
K
,
L
(
r
,
s
)
=
min
h
(
⌈
r
/
h
⌉
+
⌈
s
/
h
⌉
−
1
)
h
,
where
h runs over the set of
K-dimensions of all finite-dimensional intermediate fields
K
⊂
H
⊂
L
. This bound is closely related to one appearing in additive number theory.</description><identifier>ISSN: 0022-314X</identifier><identifier>EISSN: 1096-1658</identifier><identifier>DOI: 10.1016/j.jnt.2008.06.004</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>Additive number theory ; Combinatorics ; Commutative field extension ; Mathematics ; Number Theory ; Product set</subject><ispartof>Journal of number theory, 2009-02, Vol.129 (2), p.339-348</ispartof><rights>2008 Elsevier Inc.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c374t-e2ac66d7d8cf693aa015d02703e335fc3385a5fdffa10b008bbd62878b9b4b6e3</citedby><cites>FETCH-LOGICAL-c374t-e2ac66d7d8cf693aa015d02703e335fc3385a5fdffa10b008bbd62878b9b4b6e3</cites><orcidid>0000-0002-2305-6205</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,780,784,885,27923,27924</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00259373$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Eliahou, Shalom</creatorcontrib><creatorcontrib>Kervaire, Michel</creatorcontrib><creatorcontrib>Lecouvey, Cédric</creatorcontrib><title>On the product of vector spaces in a commutative field extension</title><title>Journal of number theory</title><description>Let
K
⊂
L
be a commutative field extension. Given
K-subspaces
A
,
B
of
L, we consider the subspace
〈
A
B
〉
spanned by the product set
A
B
=
{
a
b
|
a
∈
A
,
b
∈
B
}
. If
dim
K
A
=
r
and
dim
K
B
=
s
, how small can the dimension of
〈
A
B
〉
be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on
dim
K
〈
A
B
〉
turns out, in this case, to be provided by the numerical function
κ
K
,
L
(
r
,
s
)
=
min
h
(
⌈
r
/
h
⌉
+
⌈
s
/
h
⌉
−
1
)
h
,
where
h runs over the set of
K-dimensions of all finite-dimensional intermediate fields
K
⊂
H
⊂
L
. This bound is closely related to one appearing in additive number theory.</description><subject>Additive number theory</subject><subject>Combinatorics</subject><subject>Commutative field extension</subject><subject>Mathematics</subject><subject>Number Theory</subject><subject>Product set</subject><issn>0022-314X</issn><issn>1096-1658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLAzEUhYMoWB8_wF22Lma8mUwyM7ixFLVCoRsFdyGTB83QTkqSDvrvTam4dHXhcr4D50PojkBJgPCHoRzGVFYAbQm8BKjP0IxAxwvCWXuOZgBVVVBSf16iqxgHAEJYw2boaT3itDF4H7w-qIS9xZNRyQcc91KZiN2IJVZ-tzskmdxksHVmq7H5SmaMzo836MLKbTS3v_cafbw8vy-WxWr9-raYrwpFmzoVppKKc93oVlneUSmBMA1VA9RQyqyitGWSWW2tJNDnGX2vedU2bd_1dc8NvUb3p96N3Ip9cDsZvoWXTiznK3H85YWsow2dSM6SU1YFH2Mw9g8gII66xCCyLnHUJYBntM7M44kxecTkTBBROTMqo13IQoT27h_6B79Ycm0</recordid><startdate>20090201</startdate><enddate>20090201</enddate><creator>Eliahou, Shalom</creator><creator>Kervaire, Michel</creator><creator>Lecouvey, Cédric</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0002-2305-6205</orcidid></search><sort><creationdate>20090201</creationdate><title>On the product of vector spaces in a commutative field extension</title><author>Eliahou, Shalom ; Kervaire, Michel ; Lecouvey, Cédric</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c374t-e2ac66d7d8cf693aa015d02703e335fc3385a5fdffa10b008bbd62878b9b4b6e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Additive number theory</topic><topic>Combinatorics</topic><topic>Commutative field extension</topic><topic>Mathematics</topic><topic>Number Theory</topic><topic>Product set</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Eliahou, Shalom</creatorcontrib><creatorcontrib>Kervaire, Michel</creatorcontrib><creatorcontrib>Lecouvey, Cédric</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Journal of number theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Eliahou, Shalom</au><au>Kervaire, Michel</au><au>Lecouvey, Cédric</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the product of vector spaces in a commutative field extension</atitle><jtitle>Journal of number theory</jtitle><date>2009-02-01</date><risdate>2009</risdate><volume>129</volume><issue>2</issue><spage>339</spage><epage>348</epage><pages>339-348</pages><issn>0022-314X</issn><eissn>1096-1658</eissn><abstract>Let
K
⊂
L
be a commutative field extension. Given
K-subspaces
A
,
B
of
L, we consider the subspace
〈
A
B
〉
spanned by the product set
A
B
=
{
a
b
|
a
∈
A
,
b
∈
B
}
. If
dim
K
A
=
r
and
dim
K
B
=
s
, how small can the dimension of
〈
A
B
〉
be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on
dim
K
〈
A
B
〉
turns out, in this case, to be provided by the numerical function
κ
K
,
L
(
r
,
s
)
=
min
h
(
⌈
r
/
h
⌉
+
⌈
s
/
h
⌉
−
1
)
h
,
where
h runs over the set of
K-dimensions of all finite-dimensional intermediate fields
K
⊂
H
⊂
L
. This bound is closely related to one appearing in additive number theory.</abstract><pub>Elsevier Inc</pub><doi>10.1016/j.jnt.2008.06.004</doi><tpages>10</tpages><orcidid>https://orcid.org/0000-0002-2305-6205</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-314X |
| ispartof | Journal of number theory, 2009-02, Vol.129 (2), p.339-348 |
| issn | 0022-314X 1096-1658 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_00259373v1 |
| source | Elsevier:Jisc Collections:Elsevier Read and Publish Agreement 2022-2024:Freedom Collection (Reading list) |
| subjects | Additive number theory Combinatorics Commutative field extension Mathematics Number Theory Product set |
| title | On the product of vector spaces in a commutative field extension |
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