The Woods Hole Oceanographic Institution is a private, nonprofit research and higher education facility dedicated to the study of marine science and engineering. Its agenda includes: geological activity deep within the earth. Established in 1930 in Woods Hole, Massachusetts, it is the largest independent oceanographic research institution in the U. S. with staff and students numbering about 1,000. The Institution is organized into six departments, the Cooperative Institute for Climate and Ocean Research, a marine policy center, its shore-based facilities are located in the village of Woods Hole, United States and a mile and a half away on the Quissett Campus. The bulk of the Institution's funding comes from grants and contracts from the National Science Foundation and other government agencies, augmented by foundations and private donations. WHOI scientists and students collaborate to develop theories, test ideas, build seagoing instruments, collect data in diverse marine environments. Ships operated by WHOI carry research scientists throughout the world’s oceans.

The WHOI fleet includes two large research vessels, the coastal craft Tioga, small research craft such as the dive-operation work boat Echo, the deep-diving human-occupied submersible Alvin, the tethered, remotely operated vehicle Jason/Medea, autonomous underwater vehicles such as the REMUS and SeaBED. WHOI offers post-doctoral studies in marine science. There are several fellowship and training programs, graduate degrees are awarded through a joint program with the Massachusetts Institute of Technology. WHOI is accredited by the New England Association of Colleges. WHOI offers public outreach programs and informal education through its Exhibit Center and summer tours; the Institution has a membership program, WHOI Associate. On October 1, 2015, Mark R. Abbott became director. In 1927, a National Academy of Sciences committee concluded that it was time to "consider the share of the United States of America in a worldwide program of oceanographic research." The committee's recommendation for establishing a permanent independent research laboratory on the East Coast to "prosecute oceanography in all its branches" led to the founding in 1930 of the Woods Hole Oceanographic Institution.

A $2.5 million grant from the Rockefeller Foundation supported the summer work of a dozen scientists, construction of a laboratory building and commissioning of a research vessel, the 142-foot ketch Atlantis, whose profile still forms the Institution's logo. WHOI grew to support significant defense-related research during World War II, began a steady growth in staff, research fleet, scientific stature. In 1977 the institute appointed the influential oceanographer John Steele as director, he served until his retirement in 1989. On 1 September 1985, a joint French-American expedition led by Jean-Louis Michel of IFREMER and Robert Ballard of the Woods Hole Oceanographic Institution identified the location of the wreck of the RMS Titanic which sank off the coast of Newfoundland 15 April 1912. On 3 April 2011, within a week of resuming of the search operation for Air France Flight 447, a team led by WHOI, operating full ocean depth autonomous underwater vehicles owned by the Waitt Institute discovered, by means of sidescan sonar, a large portion of debris field from flight AF447.

In March 2017 the institution effected an open-access policy to make its research publicly accessible online. The Institution has maintained a long and controversial business collaboration with the treasure hunter company Odyssey Marine. WHOI has participated in the location of the San José galleon in Colombia for the commercial exploitation of the shipwreck by the Government of President Santos and a private company. In 2019, iDefense reported that China's hackers had launched cyberattacks on dozens of academic institutions in an attempt to gain information on technology being developed for the United States Navy; some of the targets included the Woods Hole Oceanographic Institution. The attacks have been underway since at least April 2017. In popular culture, The Woods Hole Oceanographic Institution is mentioned by name in the 2nd paragraph of chapter 13 of Peter Benchley’s famous 1973 novel “Jaws”. In the book, Quint objects to having a shark cage aboard his F/V Orca; the cage was ordered by Matt Hooper to be brought to the dock.

The shark cage was brought by a man in a pickup truck with the words “Woods Hole Oceanographic Institute” on the side. The B. H. Ketchum award, established in 1983, is presented for innovative coastal/nearshore research and is named in honor of oceanographer Bostwick H. "Buck" Ketchum. The award is administered by Rinehart Coastal Research Center. Recipients: 2017: Don Anderson, Woods Hole Oceanographic Institution 2015: Candace Oviatt, Graduate School of Oceanography, University of Rhode Island 2010: James E. Cloern, United States Geological Survey 2007: Richard Garvine, University of Delaware 2003: John Farrington, Woods Hole Oceanographic Institution 2003: Nancy Rabalais, Louisiana Universities Marine Consortium 1999: Willard Moore, University of South Carolina 1996: Ronald Smith, Loughbororugh University 1995: Christopher Martens, University of North Carolina 1992: Scott Nixon, University of Rhode Island 1990: Daniel Lynch, Dartmouth College 1989: William Boicourt, University of Maryland 1988: Alasdair McIntyre, Aberdeen University 1986: John S. Allen, Orego

In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form ×. The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example, 5 = e 2 π i / 5 − e 4 π i / 5 − e 6 π i / 5 + e 8 π i / 5, − 3 = e 2 π i / 3 − e 4 π i / 3, 3 = e 2 π i / 12 − e 10 π i / 12; the theorem is named after Heinrich Martin Weber. The Kronecker–Weber theorem can be stated in terms of fields and field extensions; the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over Q, an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.

For a given abelian extension K of Q there is a minimal cyclotomic field. The theorem allows one to define the conductor of K as the smallest integer n such that K lies inside the field generated by the n-th roots of unity. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact generalised in class field theory; the theorem was first stated by Kronecker though his argument was not complete for extensions of degree a power of 2. Weber published a proof, but this had some gaps and errors that were pointed out and corrected by Neumann; the first complete proof was given by Hilbert. Lubin and Tate proved the local Kronecker–Weber theorem which states that any abelian extension of a local field can be constructed using cyclotomic extensions and Lubin–Tate extensions. Hazewinkel and Lubin gave other proofs. Hilbert's twelfth problem asks for generalizations of the Kronecker–Weber theorem to base fields other than the rational numbers, asks for the analogues of the roots of unity for those fields.

Ghate, Eknath, "The Kronecker-Weber theorem", in Adhikari, S. D.. "An Elementary Proof of the Kronecker-Weber Theorem". American Mathematical Monthly. 81: 601–607. Doi:10.2307/2319208. JSTOR 2319208. Hazewinkel, Michiel, "Local class field theory is easy", Advances in Mathematics, 18: 148–181, doi:10.1016/0001-870890156-5, ISSN 0001-8708, MR 0389858 Hilbert, David, "Ein neuer Beweis des Kronecker'schen Fundamentalsatzes über Abel'sche Zahlkörper.", Nachrichten der Gesellschaft der Wissenschaften zu Göttingen: 29–39 Kronecker, Leopold, "Über die algebraisch auflösbaren Gleichungen", Berlin K. Akad. Wiss.: 365–374, ISBN 9780821849828, Collected works volume 4 Kronecker, Leopold, "Über Abelsche Gleichungen", Berlin K. Akad. Wiss.: 845–851, ISBN 9780821849828, Collected works volume 4 Lemmermeyer, Franz, "Kronecker-Weber via Stickelberger", Journal de théorie des nombres de Bordeaux, 17: 555–558, arXiv:1108.5671, doi:10.5802/jtnb.507, ISSN 1246-7405, MR 2211307 Lubin, Jonathan, "The local Kronecker-Weber theorem", Transactions of the American Mathematical Society, 267: 133–138, doi:10.2307/1998574, ISSN 0002-9947, JSTOR 1998574, MR 0621978 Lubin, Jonathan.

A new proof of the Kronecker-Weber theorem, Trudy Mat. Inst. Steklov. 38, Moscow: Izdat. Akad. Nauk SSSR, pp. 382–387, MR 0049233 Schappacher, Norbert, "On the history of Hilbert's twelfth problem: a comedy of errors", Matériaux pour l'histoire des mathématiques au XXe siècle, Sémin. Congr. 3, Paris: Société Mathématique de France, pp. 243–273, ISBN 978-2-8

Rear Admiral Keith McNeil Campbell-Walter CB was a senior Royal Navy officer. Born on 31 August 1904, Keith McNeil Campbell-Walter was educated at Bedford School and at Britannia Royal Naval College, he was served during the Second World War. He was promoted to the rank of Commander in 1938, Captain in 1945, Commodore in 1954 and Rear Admiral in 1955, he was appointed Aide-de-camp to Queen Elizabeth II in 1954, Flag Officer and Commander of Allied Naval Forces Northern Area, Central Europe, between 1955 and 1958. Rear Admiral Keith McNeil Campbell-Walter was invested as a Companion of the Order of the Bath in 1957, he retired from the Royal Navy in 1958 and died on 24 April 1976, aged 71. He is the grandfather of Baroness Francesca Thyssen-Bornemisza and Jamie Campbell-Walter